"Finding Probability Using Enumeration Method" PPT courseware

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"Finding Probability Using Enumeration Method" PPT courseware

Part One: Course Review

No matter what event probability we are looking for, we can do a lot of experiments. Finding the frequency to get the probability is the content of the previous lesson. It is universal, but it is really troublesome to find. Is there a simpler method? This method is the method we are going to introduce today - the enumeration method.

review:

1. Randomly draw one of the 5 paper lots marked with numbers 1, 2, 3, 4, and 5. The number on the drawn lot has 5 possible results, namely 1, 2, 3, 4, 5. The probability of drawing each lot is equal, which is 1/5.

2. Roll a dice. There are 6 possible outcomes for the number on the upward side, namely 1, 2, 3, 4, 5, and 6. The probability of each point appearing is equal, which is 1/6.

(1) What do the above two experiments have in common?

There are only a limited number of possible outcomes in an experiment. In an experiment, all outcomes are equally likely to occur.

(2) For experiments with classical concepts, how to find the probability of an event?

Generally speaking, if there are n possible outcomes in an experiment, and they are all equally likely to occur, and event A contains m of the outcomes, then the probability of event A occurring is P(A)=m/n.

Use enumeration method to find probability PPT, part 2 content: knowledge points

Generally speaking, if there are n possible outcomes in an experiment, and they are all equally likely to occur, and event A contains m of the outcomes, then the probability of event A occurring is P(A)=m/n.

In the probability formula P(A)=m/n, what are the values ​​of m and n, the quantitative relationship between m and n, and the value range of P(A).

0 ≤ m ≤ n, m and n are natural numbers

∵0 ≤m/n≤ 1, ∴0≤P(A) ≤1.

When m=n, A is an inevitable event, probability P(A)=1,

When m=0, A is an impossible event, and the probability P(A)=0.

Using enumeration method to find probability PPT, part three: analysis of examples

Example 1: Throw a cube dice with uniform texture and observe the number on the upward side.

(1) Find the probability that the number of throws is 2 or 4 or 6;

(2) When Xiao Ming was doing the experiment of throwing dice, he did not get a point of 2 in the first five throws. Find the probability that he got a point of 2 in the sixth throw.

Solution: Roll a cube dice with uniform texture. The points on the upward side may be 1, 2, 3, 4, 5, 6, a total of 6 types. These points are equally likely to occur.

(1) There are three outcomes when the number of rolls is 2, 4, or 6 (recorded as event A), so P(A)=3/6=1/2;

(2) Xiao Ming did not get the number 2 in the first five throws, but the number he got in the sixth throw could still be 1, 2, 3, 4, 5, 6, a total of 6 ways. His sixth roll of 2 (recorded as event B) has 1 outcome, so P(B)=1/6.

Example 2: There are four small balls in the pocket, one red and three black. Two small balls are taken out at a time. Find the probability that "all the small balls taken out are black balls"

List directly

Solution: When two balls are taken out of the pocket at one time, there are 6 possible outcomes, namely (red, black 1) (red, black 2) (red, black 3) (black 1, black 2) (black 1, black 3)

(Black 2, Black 3) and they are equally likely to appear. There are three results that satisfy that all the balls taken out are black balls (recorded as event A), namely (black 1, black 2) (black 1, black 3) (black 2, black 3), then P (A) = 3/6 =1/2

Use enumeration method to find probability PPT, part 4 content: Think about it

Are all the possible outcomes of the two experiments "tossing two coins at the same time" and "tossing a coin twice in succession" the same?

The result is the same, but there is sometimes a difference between tossing two coins at the same time and tossing a coin twice in succession. For example, when throwing in succession, there will be such a question: What is the probability of heads appearing first and then tails? This has to do with sequence. This problem does not arise when two coins are tossed simultaneously.

Use enumeration method to find probability PPT, part 5: tree diagram method

Step 1: The possible results are A and B. They are equally likely to occur and are written in no particular order. They are written on the first line.

Step 2: The possible results are C, D and E. The three are equally likely to occur and are not in order. Draw three branches from A and B respectively. Write C, D and E respectively on the second line under the branches. D and E.

Step 3: There are two possible results, H and I. They are equally likely to occur and are not listed in order. Draw two branches from C, D and E, and write on the third line under the branch. H and I. (If there are more steps, you can continue above)

Step 4: Write the possible results vertically below to get the total number of all possible results. Then find the number of species that meet the requirements, and you can use probability and meaning to calculate the probability.

Using enumeration method to find probability PPT, Part 6: Class Summary

1. Generally speaking, if there are n possible outcomes in an experiment, and they are all equally likely to occur, and event A contains m of the outcomes, then the probability of event A occurring is P(A)=m/ n.

2. The steps to use the list method to find probability are as follows:

①List;

② Determine the values ​​of m and n in the formula P(A)=m/n by counting in the table;

③Use the formula P(A)=m/n to calculate the probability of the event.

3. The steps to use the tree diagram method to find probability are as follows:

①Draw a tree diagram;

② List the results and determine the values ​​of m and n in the formula P(A)=m/n;

③Use the formula P(A)=m/n to calculate the event probability.

4. When an event involves two factors and the number of possible outcomes is large, the list method is usually used. When an experiment involves three or more factors, "drawing a tree diagram" is usually used.

Use enumeration method to find probability PPT, Part 7: Consolidation Practice

1. Roll a die. There are ____ possibilities for the number on the up side. The probability of each possibility is ____.

2. There are 2 white balls and 1 black ball in the pocket. If you pick any ball from them, the probability of getting the white ball is ______. The probability of getting the black ball is ____.

3. From a deck of fully and evenly mixed playing cards, randomly select a card. The probability of drawing a king is ( ), the probability of drawing a 6 is ( ), and the probability of drawing a spade is ( ).

4. Draw circles, parallelograms, equilateral triangles, and squares on four cards of the same shape, size, and texture respectively, then place them on the table, shuffle them, and randomly select one card. The probability of drawing an axially symmetrical figure is ( ), the probability of drawing a centrally symmetrical figure is ( ).

5. There are four seats next to a round table. A sits on the seat as shown in the picture first, and B, C, and D randomly sit on the other three seats. Find the probability that A and B are not sitting next to each other.

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