A calculator that allows you to calculate possible probabilities. Classic formula for calculating probability

Example 1. The project includes the following works presented in the table. Build a network schedule for completing a set of works.

Solution. Works a 1 and a 2 are not preceded by any work, therefore, on the graph they are depicted as arcs emerging from original event(1), which means the moment the project begins. Work a 3 preceded by work a 1, so on the graph there is an arc a 3 directly follows the arc a 1 . Event (2) means the moment of completion of work a 1 and the beginning of the work that it precedes. Work a 4 preceded by work a 1 and a 2. This dependence is reflected in the graph by introducing fictitious work (2, 3). The moment of the event (3) will be the moment by which the work will be completed a 1 and a 2 and work can begin a 4 . Similarly, taking into account the relationships, all other work is depicted on the graph. The final event (6) means the moment the entire project is completed.

Rules used when constructing a network diagram.

1) there should be no “dead ends” in network diagrams, i.e. events from which no work comes out (except for the termination event);

2) in network graphs there should be no events (except for outgoing ones) that are not preceded by at least one job;

3) when constructing network graphs, it is impossible to allow two adjacent events to be connected by two or a large number the number of works, which most often happens when depicting works performed in parallel. This error causes confusion because two various works will have the same designation. To avoid this, it is recommended to introduce additional events and associate it with a subsequent dependency or dummy job;

4) there should be no closed loops in the network, i.e. chains connecting certain events to themselves;

5) in addition, if any complex work can be started before complete completion immediately preceding work, the latter is depicted as a series of sequentially performed works, each of which ends with a specific event.

6) if to perform one of the works it is necessary to obtain the results of all the works included in the event preceding it, and for another work it is enough to obtain the result of only one or several of these works, then a new event must be additionally introduced, as well as a fictitious work connecting a new event from the previous one.

A schedule constructed in compliance with these rules is a network model of project execution. In this case, first, private network diagrams are usually drawn up, covering work on individual, independent meaning parts of the overall set of works, and then by “stitching together” a comprehensive (consolidated) schedule is obtained, covering the entire set of works to be performed.

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Network diagrams and rules for their construction

A network diagram is a graphical representation of the processes that are necessary to achieve a goal.

Network planning and control methods (NPM) are based on graph theory. A graph is a collection of two finite sets: a set of points, called vertices, and a set of pairs of vertices, called edges. There are two types of graphs commonly used in economics: tree and network. A tree is a connected graph without cycles, having an initial vertex (root) and extreme vertices. A network is a directed finite connected graph that has a starting vertex (source) and an ending vertex (sink). Thus, each network graph is a network consisting of nodes (vertices) and oriented arcs (edges) connecting them. The graph nodes are called events, and the oriented arcs connecting them are called jobs. On a network diagram, events are depicted by circles or other geometric shapes, and the work connecting them with dimensionless arrows (they are called dimensionless because the length of the arrow does not depend on the amount of work that it reflects).

Each event in the network diagram is assigned a specific number ( i), and the work connecting events is denoted by an index ( ij). Each job is characterized by its duration (duration) t(ij). Meaning t(ij) in hours or days is indicated as a number above the corresponding arrow on the network diagram.

In network planning practice, several types of work are used:

1) real work, a production process that requires labor, time, materials;

2) passive work (waiting), a natural process that does not require labor or material resources, but the implementation of which can only occur during certain period time;

3) fictitious work (dependence), which does not require any costs, but shows that some event cannot happen before another. When constructing a schedule, such work is usually indicated by a dotted line.

Each work, alone or in combination with other works, ends with events that express the results of the work performed. In network graphs, the following events are distinguished: 1) initial, 2) intermediate, 3) final (final). If the event is of an intermediate nature, then it is a prerequisite for the start of the work that follows it. It is considered that the event has no duration and occurs instantly after the completion of the work preceding it. The initial event is not preceded by any work. It expresses the moment of occurrence of conditions for the start of the entire complex of work. The final event does not have any subsequent work and expresses the moment of completion of the entire complex of work and achievement of the intended goal.

Interconnected activities and events of the network diagram form paths that connect the initial and final events; they are called complete. Full path on a network diagram represents a sequence of activities in the direction of the arrows from the initial to the final event. The complete path of maximum duration is called critical. The duration of the critical path determines the final deadline for completing the entire set of works and achieving the intended goal.

Activities located on the critical path are called critical or stressful. All other work is considered non-critical (non-stressful) and has time reserves that allow you to move the deadlines for their completion and the timing of events without affecting the overall duration of the entire complex of work.

Rules for constructing a network diagram.

1. The network is drawn from left to right, and each event with a large serial number is depicted to the right of the previous one. General direction The arrows representing the jobs should also generally be arranged from left to right, with each job leaving the lower numbered event and entering the higher numbered event.

Incorrect Correct

3. There should be no “dead ends” in the network, that is, all events except the final one must have subsequent work (dead ends are intermediate events from which no work comes out). This situation may occur when this work is not needed or any work is missing.

4. There should be no events in the network, other than the initial one, that are not preceded by at least one activity. Such events are called “tail” events. This may occur if previous work is missed.

To correctly number network diagram events, use the following action scheme. Numbering begins from the initial event, which is assigned the number 0 or 1. From the initial event (1), all works emanating from it (oriented arcs) are crossed out, and on the remaining network an event is again found, which does not include any work. This event is assigned a number (2). The specified sequence of actions is repeated until all network diagram events are numbered. If, during the next deletion, two events occur simultaneously that have no incoming work, then numbers are assigned to them randomly. The ending event number must be equal to the quantity events in the network diagram.

Example.

In the process of constructing a network schedule, it is important to determine the duration of each work, that is, it is necessary to give it a time estimate. The duration of work is established either in accordance with current standards or on the basis expert assessments. In the first case, duration estimates are called deterministic, in the second - stochastic.

Exist various options calculation of stochastic time estimates. Let's look at some of them. In the first case, three types of duration of specific work are established:

1) the maximum period, which comes from the most not favorable conditions doing the work ( t max);

2) the minimum period, which is based on the most favorable conditions for performing the work ( tmin);

3) the most probable period, based on the actual availability of resources for the work and the presence of normal conditions for its implementation ( t in).

Based on these estimates, the expected time to complete the work (its time estimate) is calculated using the formula

. (5.1)

In the second case, two estimates are given - the minimum ( tmin) and maximum ( t max). The duration of work in this case is considered as random value, which as a result of implementation can take any value in a given interval. Expected value of these estimates ( t cool) (with a beta probability density distribution) is estimated by the formula

. (5.2)

To characterize the degree of dispersion of possible values ​​around the expected level, the dispersion indicator ( S 2)

. (5.3)

The construction of any network diagram begins with drawing up full list works Then the order of work is established, and for each specific work the immediately preceding and subsequent work is determined. To establish the boundaries of each type of work, the following questions are used: 1) what should precede this work and 2) what should follow this work. After compiling a complete list of works, establishing their priority and time estimates, they proceed directly to the development and preparation of a network schedule.

Example.

Consider, as an example, the construction program for a warehouse building. The list of operations, their sequence and time duration will be presented in a table.

Table 5.1

List of network diagram works

Built on the basis of the data in table. 5.1, the preliminary network schedule for the work is as follows (Fig. 5.1).

Rice. 5.1. Preliminary network schedule

Below is the same schedule for the construction of a warehouse building, numbered and with time estimates for the work indicated (Fig. 5.2).

Rice. 5.2. Final version network graphics

A local network or LAN is two or more computers connected to each other directly or through a router (router) and capable of exchanging data. Such networks usually cover a small office or home space and are used to share an Internet connection, as well as for other purposes such as file sharing or online gaming. In this article we will talk about how to build a local network of two computers.

As it becomes clear from the introduction, you can combine two PCs into a local area in two ways - directly, using a cable, and through a router. Both of these options have their pros and cons. Below we will look at them in more detail and learn how to configure the system for data exchange and access to the Internet.

Option 1: Direct connection

With this connection, one of the computers acts as a gateway for connecting to the Internet. This means that it must have at least two network ports. One for the global network, and the second for the local one. However, if the Internet is not required or it “comes” without the use of wires, for example, through a 3G modem, then you can get by with one LAN port.

The connection diagram is simple: the cable is plugged into the corresponding connectors on the motherboard or network card of both machines.

Please note that for our purposes we need a cable (patch cord) that is designed for direct connection of computers. This type is called a “crossover”. However, modern equipment is capable of independently determining pairs for receiving and transmitting data, so a regular patch cord will most likely also work normally. If problems arise, you will have to redo the cable or find the right one in the store, which can be very difficult.

The advantages of this option include ease of connection and minimum requirements by equipment. Actually, we only need a patch cord and LAN card, which in most cases is already built into motherboard. The second plus is high data transfer speed, but this depends on the capabilities of the card.

It's a stretch to call the disadvantages such - they reset the settings when reinstalling the system, as well as the inability to access the Internet when the PC, which is the gateway, is turned off.

Settings

After connecting the cable, you need to configure the network on both PCs. First, you need to assign each machine in our local area unique name. This is necessary in order to software could find computers.

Now you need to configure resource sharing in local network, since it is limited by default. These steps also need to be performed on all machines.

1. Right-click on the connection icon in the notification area and open "Network and Internet settings".

   

2. Let's move on to setting up sharing parameters.

   

3. For private network(see screenshot) allow discovery, enable file and printer sharing, and let Windows manage connections.

   

4. For the guest network we also enable discovery and sharing.

   

5. For all networks, we disable general access, configure encryption with 128-bit keys, and disable password access.

   

6. Save the settings.

   

In Windows 7 and 8, this block of parameters can be found like this:

1. On the first PC (the one that connects to the Internet), after going to the parameters (see above), click on the menu item "Configuring adapter settings".

   

2. Here we select "LAN connection", right-click on it and go to properties.

   

3. In the list of components we find the protocol IPv4 and, in turn, move on to its properties.

   

4. Switch to manual input and in the field "IP address" enter the following numbers:

   In field "Subnet mask" the required values ​​will be automatically entered. There is no need to change anything here. This completes the setup. Click OK.

   

5. On the second computer, you must set the following IP address in the protocol properties:

   We leave the mask as default, but in the fields for the gateway and DNS server addresses we indicate the IP of the first PC and click OK.

   

   In the "seven" and "eight" you should go to "Network Control Center" from the notification area, and then click on the link "Change adapter settings". Further manipulations are carried out according to the same scenario.

   

The final procedure is to allow shared access to the Internet.

Now on the second machine it will be possible to work not only on the local network, but also on the global one. If you need to exchange data between computers, you will need to perform one more setup, but we will talk about this separately.

Option 2: Connection via router

For such a connection, we will need, in fact, the router itself, a set of cables and, of course, the corresponding ports on the computers. The type of cables for connecting machines to a router can be called “direct”, as opposed to a crossover cable, that is, the cores in such a wire are connected “as is”, directly (see above). Such wires with already mounted connectors can be easily found in retail stores.

The router has several connection ports. One for receiving the Internet and several for connecting computers. It’s easy to distinguish them: LAN connectors (for cars) are grouped by color and numbered, and the port for the incoming signal stands apart and has a corresponding name, usually written on the case. The connection diagram in this case is also quite simple - the cable from the provider or modem is connected to the connector "Internet" or, in some models, "Link" or "ADSL", and computers to ports signed as "LAN" or "Ethernet".

The advantages of this scheme are the ability to organize a wireless network and automatically determine system parameters.

The following concepts and terminology are adopted in the system of network planning and construction production management.

The concept of a project summarizes the range of organizational and technical problems, solved to achieve the final results of construction production. These include: development of a feasibility study for the planned construction, selection of a construction site, carrying out engineering and geological surveys, registration of the territory for development, development and approval of technical documentation necessary for construction, including schedules and diagrams of construction and installation work prior to delivery of the constructed objects into operation.

A set of works performed to achieve specific purpose, which determines a certain part of the project, is called the project function. For example, work related to the preparation of construction production (development of working drawings of buildings and structures, work design; placing orders for the manufacture of equipment, structures and their delivery to the construction site, etc.) or with the production of construction and installation works, with the construction foundations (construction of stripping, laying out axes, digging pits, preparing and installing formwork and reinforcement, preparing concrete mixture, transporting and laying it in formwork, stripping and capturing the cavities of concrete foundations with soil) are functions in the design of the structure.

The most important indicators of project efficiency are the cost and duration of construction, which are directly dependent on similar indicators of individual project functions. If a list of all project functions has been established and the sequence of execution and time spent on each of them have been determined, then by depicting these functions in the form of a graphical network, you can see which of them determine the deadlines for the remaining functions and the entire project as a whole.

It follows that the network diagram reflects the logical relationship and interdependence of all organizational, technical and production operations for the implementation of the project, as well as a certain sequence of their implementation.

The main parameters of a network diagram are work and event, and its derivatives are network, critical path and slack.

Work means any process that requires time. In network diagrams, this term determines not only certain production processes that require the expenditure of material resources, but also the expected processes associated with observing technological breaks, for example, for hardening laid concrete.

An event is an intermediate or final result of one or more activities, necessary for the start of other activities. The event occurs after all the work included in it has been completed. Moreover, the moment of completion of the event is the moment of completion of the last (included in it work. Thus, the event is final results of certain works and at the same time - starting positions for the beginning of subsequent ones. An event that has no previous activities is called initial; an event that has no subsequent activities is called final.

The work on the network diagram is represented by one solid arrow. The duration of the work in time units (days, weeks) is indicated under the arrow, and the name of the work above the arrow. Each event is depicted by a circle and numbered (Fig. 115).

Rice. 115. Designation of events and work m - n.

Rice. 116. Designation of the dependence of technological events.

Rice. 117. Designation of the dependence of events of an organizational nature.

The duration of a particular work, established depending on the accepted method of its implementation according to Unified Research Work or calculations of labor costs, is called a time estimate. A dependency between individual events that does not require time or resources is called dummy work and is represented by a dotted arrow on a network diagram.

These dependencies or fictitious work can be divided into three groups: technological, organizational, conditional.

Technological dependence means that the completion of one job depends on the completion of another, for example, the walls of the next floor cannot be laid before the floor panels of the lower floor are installed (Fig. 116).

The dependence of an organizational nature shows the transitions of teams of workers, the transfer of mechanisms from one site to another, etc. They arise mainly when performing work using continuous methods (Fig. 117).

If there are several final events (for example, the commissioning of several objects included in the start-up complex of an enterprise), they should be linked by conditional dependencies or fictitious work together - the commissioning of the enterprise (Fig. 118, b).

There must be one initial event. In cases where there are several initial events (for example, work on excavation pits for several objects begins independently of each other), they should be conditionally connected by the designation of fictitious work with a single initial event (Fig. 118, a).

If the timing of the actual initial events of individual objects of the complex is different, the concept of dependencies with real time consumption, converging at one initial node, should be introduced.

The duration established taking into account single-shift work, and for leading machines two-shift work and the optimal saturation of the work front, is called normal work duration. If the duration of work is determined by the maximum workload of the work front during two or three shift work, then it is considered minimal.

Rice. 118. Designation of conditional dependencies.

The term of work differs in terms:

the earliest work start date is the first day on which work can begin;

the earliest work completion date is the day the work ends if it started at the earliest start date;

most late date start of work - the last day of the start of work without delaying the total construction period;

the latest completion date for work is the day on which the work must be completed without delaying construction, i.e. without disrupting the overall construction period.

The difference between the latest and earliest start dates for work determines the private slack, i.e., the time by which work can be postponed without increasing the duration of construction. The amount of time that a job can be deferred without delaying any subsequent work determines the total float, which is the difference between the total slack of the job in question and the subsequent job. In the case of several subsequent jobs, the job that has the smallest total float time is selected.

A continuous sequence of works and events from initial to final, requiring the greatest time for its completion, determines the critical path, which determines the total duration of construction, since those lying on it critical works have no time reserves.

In network diagrams, the direction of the arrows depicting work can be chosen arbitrarily. Typically, such graphs are plotted from left to right. However, the arrows for individual job types can go up, down, or from right to left.

When drawing up a network schedule, each job should be considered from the point of view of its connection with other jobs and answer the following questions:

what work should be completed before starting this work;

what other work can be completed simultaneously with this work;

what work cannot be started until the work is completed. Let's look at some examples graphic image connections and sequence of work in network diagrams.

Rice. 119. Schemes of connection between works (a, b, c, d, e, f, g - cases 1,2,3,4,5,6,7).

Case 1 (Fig. 119, a). Dependence between jobs A (1-2) and B (2-3). Job B cannot start until job A is finished.

Case 2 (Fig. 119.6). Dependence of two jobs on one. Works D (7-8) and E (7-9) cannot be started until work D (6-7) is completed.

Case 3 (Fig. 119, c). Dependence of one job on the completion of two jobs. Work E (10-11) cannot begin until work D (8-10) and D (9-10) are completed.

Case 4 (Fig. 119, d). The beginning of two jobs depends on the completion of two jobs. Works E (15-16) and D (15-17) can begin only after completion of works B (13-15) and C (14-15).

Case 5 (Fig. 119, 6). Dependence of two groups of work. Work B (15-16) depends only on the completion of work A (14-15), and work D (21-22) depends on the completion of work A (14-45) and B (19-21). The network is linked by enabling fictitious work D (15-21).

Case 6 (Fig. 119, f). Work D (47-48) cannot be started until work B (46-47) is completed. In turn, work B (50-51) cannot be started until work B (46-47) and A (49-50) are completed. Work E (47-50) is fictitious, defining the logical connection of the network by delaying the start of work B (50-51) until work B (46-47) is completed.

Case 7 (Fig. 119,g). Work D (8-14) cannot be started until work A (2-8) and B (4-6) are completed; work G (12-16) cannot be started until the completion of Fig. 120. Network diagram, works D (10-12), B (4-6); the relationship between these jobs is indicated by the fictitious job E (6-12). Since work J (12-16) does not depend on the completion of work A (2-8), it is separated from the last fictitious work B (6-8).

Rice. 120. Network diagram.

In order to understand the methodology for constructing network graphs, let us consider the case when the following conditions arose during the construction of an object:

at the beginning of construction, work A and B must be carried out in parallel;

work B, D and D can be started before work A is completed;

work B must be completed before work E and G begin;

Moreover, work E also depends on the completion of work A;

work 3 cannot be started until work D and E are completed;

work I depends on the completion of works G and 3;

work K follows the end of work J;

work A follows work K and depends on the completion of works G and 3;

the final work M depends on the completion of works B, I and L.

In Fig. 120 shows one of several possible solutions to the problem determined by the given construction conditions. All decisions must be based on the same logical concept, regardless of the grid type. The grid must be considered from the point of view of the logical sequence of work. For this purpose, its review should begin with last event on the object and go back from event to event, checking the following provisions: whether each work starting at the event depends on all the works leading to the event; whether all activities on which the activity in question must depend are included in the event. If both questions can be answered positively, then the network diagram satisfies the requirements of the designed construction technology of the facility.

When constructing a network diagram, the concept of “work”, depending on the degree of desired accuracy, can mean individual species works or complexes of production processes performed at this facility by one of the organizations participating in the construction. For example, the chief engineer of a trust needs to know less details than the manufacturer of the work. Therefore, to ensure construction management at the trust level, the network schedule can be drawn up on the basis of more aggregated indicators.

Do you want to know what mathematical odds on the success of your bet? Then there is two good news for you. First: to calculate cross-country ability, you don’t need to carry out complex calculations and spend a large number of time. It is enough to use simple formulas, which will take a couple of minutes to work with. Second: after reading this article, you can easily calculate the probability of any of your transactions passing.

To correctly determine cross-country ability, you need to take three steps:

• Calculate the percentage of probability of the outcome of an event according to the bookmaker’s office;
• Calculate the probability using statistical data yourself;
• Find out the value of the bet, taking into account both probabilities.

Let's look at each of the steps in detail, using not only formulas, but also examples.

Fast passage

Calculating the probability included in bookmaker odds

The first step is to find out with what probability the bookmaker himself estimates the chances of a particular outcome. It’s clear that bookmakers don’t set odds just like that. To do this we use the following formula:

PB=(1/K)*100%,

where P B is the probability of the outcome according to the bookmaker’s office;

K – bookmaker odds for the outcome.

Let’s say that the odds for London Arsenal’s victory in the match against Bayern Munich are 4. This means that the probability of their victory is assessed by the bookmaker as (1/4)*100%=25%. Or Djokovic plays against Youzhny. The multiplier for Novak's victory is 1.2, his chances are (1/1.2)*100%=83%.

This is how the bookmaker itself evaluates the chances of success of each player and team. Having completed the first step, we move on to the second.

Calculation of the probability of an event by the player

The second point of our plan is our own assessment of the probability of the event. Since we cannot mathematically take into account such parameters as motivation and game tone, we will use a simplified model and use only statistics from previous meetings. For calculation statistical probability outcome we apply the formula:

PAND=(UM/M)*100%,

WherePAND– probability of an event according to the player;

UM – the number of successful matches in which such an event occurred;

M – total matches.

To make it clearer, let's give examples. Andy Murray and Rafael Nadal played 14 matches between themselves. In 6 of them the total was less than 21 in games, in 8 the total was more. You need to find out the probability that the next match will be played with a higher total: (8/14)*100=57%. Valencia played 74 matches against Atlético at Mestalla, in which they won 29 victories. Probability of Valencia winning: (29/74)*100%=39%.

And we learn all this only thanks to the statistics of previous games! Naturally, for some new team or a player, it will not be possible to calculate such a probability, so this betting strategy is only suitable for matches in which the opponents meet more than once. Now we know how to determine the bookmaker's and our own probabilities of outcomes, and we have all the knowledge to move on to the last step.

Determining the value of a bet

The value (value) of a bet and the passability have a direct connection: the higher the value, the higher the chance of passing. The value is calculated as follows:

V=PAND*K-100%,

where V is value;

P I – probability of outcome according to the bettor;

K – bookmaker odds for the outcome.

Let’s say we want to bet on Milan’s victory in the match against Roma and we calculate that the probability of the “red-blacks” winning is 45%. The bookmaker offers us odds of 2.5 for this outcome. Would such a bet be valuable? We carry out calculations: V=45%*2.5-100%=12.5%. Great, we have a valuable bet with good chances of passing.

Let's take another case. Maria Sharapova plays against Petra Kvitova. We want to make a deal for Maria to win, the probability of which, according to our calculations, is 60%. Bookmakers offer a 1.5 multiplier for this outcome. We determine the value: V=60%*1.5-100=-10%. As you can see, this bet is of no value and should be avoided.