Question

В группе 13 студентов сделали 5 студентов прививку. К доске вышли 3 студента. Какова вероятность след событий: Все 3 привиты Только 2 Только 1 Не более 2х привитых Никто не привит Решите пожалуйста!!!

Answer 1

В группе 13 студентов сделали 5 студентов прививку.

шанс прививки у студня p=5/13. Не прививки q=1-5/13=8/13;

Какова вероятность след событий:

1) Все 3 привиты: (5/13)³=125/2197;

2)Только 2 привиты: два привитых и один нет: (5/13)²*8/13=200/2197;

3)Только 1 привит: один привит и два нет: 5/13*(8/13)²=320/2197;

4)Не более 2х привитых(0 или 1 или 2 привитых): тоже самое что не трое привитых. Полный шанс минус три привитых, что решили в первом примере: 1-125/2197=2074/2197

5)Никто не привит: 3 не привитых: (8/13)³=512/2197;

Answer 2

Problem Analysis

We are given a group of 13 students, out of which 5 students have received a vaccination. We are also told that 3 students are selected to go to the board. We need to calculate the probability of different events related to the vaccination status of these selected students.

Solution

To solve this problem, we need to consider the different possibilities for the number of vaccinated students among the 3 selected students. Let's calculate the probability for each event:

Event 1: All 3 students are vaccinated To calculate the probability that all 3 students are vaccinated, we need to consider the number of ways we can select 3 vaccinated students out of the 5 vaccinated students, divided by the total number of ways we can select any 3 students out of the 13 students.

The probability can be calculated as: ``` P(All 3 vaccinated) = (Number of ways to select 3 vaccinated students) / (Number of ways to select any 3 students) ```

Event 2: Exactly 2 students are vaccinated To calculate the probability that exactly 2 students are vaccinated, we need to consider the number of ways we can select 2 vaccinated students out of the 5 vaccinated students, multiplied by the number of ways we can select 1 non-vaccinated student out of the 8 non-vaccinated students, divided by the total number of ways we can select any 3 students out of the 13 students.

The probability can be calculated as: ``` P(Exactly 2 vaccinated) = (Number of ways to select 2 vaccinated students) * (Number of ways to select 1 non-vaccinated student) / (Number of ways to select any 3 students) ```

Event 3: Exactly 1 student is vaccinated To calculate the probability that exactly 1 student is vaccinated, we need to consider the number of ways we can select 1 vaccinated student out of the 5 vaccinated students, multiplied by the number of ways we can select 2 non-vaccinated students out of the 8 non-vaccinated students, divided by the total number of ways we can select any 3 students out of the 13 students.

The probability can be calculated as: ``` P(Exactly 1 vaccinated) = (Number of ways to select 1 vaccinated student) * (Number of ways to select 2 non-vaccinated students) / (Number of ways to select any 3 students) ```

Event 4: No more than 2 students are vaccinated To calculate the probability that no more than 2 students are vaccinated, we need to consider the sum of the probabilities of events 1, 2, and 3.

The probability can be calculated as: ``` P(No more than 2 vaccinated) = P(All 3 vaccinated) + P(Exactly 2 vaccinated) + P(Exactly 1 vaccinated) ```

Event 5: No student is vaccinated To calculate the probability that no student is vaccinated, we need to consider the number of ways we can select 3 non-vaccinated students out of the 8 non-vaccinated students, divided by the total number of ways we can select any 3 students out of the 13 students.

The probability can be calculated as: ``` P(No student vaccinated) = (Number of ways to select 3 non-vaccinated students) / (Number of ways to select any 3 students) ```

Calculation

Now, let's calculate the probabilities for each event using the given information.

Event 1: All 3 students are vaccinated The number of ways to select 3 vaccinated students out of 5 vaccinated students is given by the combination formula: ``` Number of ways to select 3 vaccinated students = C(5, 3) = 10 ``` The number of ways to select any 3 students out of 13 students is given by the combination formula: ``` Number of ways to select any 3 students = C(13, 3) = 286 ``` Therefore, the probability that all 3 students are vaccinated is: ``` P(All 3 vaccinated) = 10 / 286 ≈ 0.034965 ```

Event 2: Exactly 2 students are vaccinated The number of ways to select 2 vaccinated students out of 5 vaccinated students is given by the combination formula: ``` Number of ways to select 2 vaccinated students = C(5, 2) = 10 ``` The number of ways to select 1 non-vaccinated student out of 8 non-vaccinated students is given by the combination formula: ``` Number of ways to select 1 non-vaccinated student = C(8, 1) = 8 ``` Therefore, the probability that exactly 2 students are vaccinated is: ``` P(Exactly 2 vaccinated) = (10 * 8) / 286 ≈ 0.27972 ```

Event 3: Exactly 1 student is vaccinated The number of ways to select 1 vaccinated student out of 5 vaccinated students is given by the combination formula: ``` Number of ways to select 1 vaccinated student = C(5, 1) = 5 ``` The number of ways to select 2 non-vaccinated students out of 8 non-vaccinated students is given by the combination formula: ``` Number of ways to select 2 non-vaccinated students = C(8, 2) = 28 ``` Therefore, the probability that exactly 1 student is vaccinated is: ``` P(Exactly 1 vaccinated) = (5 * 28) / 286 ≈ 0.48951 ```

Event 4: No more than 2 students are vaccinated The probability that no more than 2 students are vaccinated is the sum of the probabilities of events 1, 2, and 3: ``` P(No more than 2 vaccinated) = P(All 3 vaccinated) + P(Exactly 2 vaccinated) + P(Exactly 1 vaccinated) ≈ 0.034965 + 0.27972 + 0.48951 ≈ 0.804195 ```

Event 5: No student is vaccinated The number of ways to select 3 non-vaccinated students out of 8 non-vaccinated students is given by the combination formula: ``` Number of ways to select 3 non-vaccinated students = C(8, 3) = 56 ``` Therefore, the probability that no student is vaccinated is: ``` P(No student vaccinated) = 56 / 286 ≈ 0.195804 ```

Summary

Based on the given information, the probabilities for the different events are as follows: - The probability that all 3 students are vaccinated is approximately 0.034965. - The probability that exactly 2 students are vaccinated is approximately 0.27972. - The probability that exactly 1 student is vaccinated is approximately 0.48951. - The probability that no more than 2 students are vaccinated is approximately 0.804195. - The probability that no student is vaccinated is approximately 0.195804.

Please let me know if you need any further assistance!