Question

Помогите пожалуйста В департаменте работают 8 юристов и 4 экономиста. Случайным образом для участия в торжественном мероприятии выбирают 3 человек. Найти вероятность того, что среди них окажутся: а) только экономисты; б) только юристы; в) ровно два юриста; г) хотя бы один экономист.

Answer 1

Problem Analysis

We are given that there are 8 lawyers and 4 economists in a department. We need to find the probability of selecting 3 people randomly for a special event, with the following conditions: a) Only economists are selected. b) Only lawyers are selected. c) Exactly two lawyers are selected. d) At least one economist is selected.

Solution

To solve this problem, we need to calculate the total number of ways to select 3 people from a group of 12 (8 lawyers + 4 economists). We can then calculate the number of ways to satisfy each condition and divide it by the total number of ways to get the probability.

Total Number of Ways

The total number of ways to select 3 people from a group of 12 is given by the combination formula:

C(n, r) = n! / (r!(n-r)!)

Where n is the total number of people and r is the number of people to be selected.

For our problem, n = 12 and r = 3. Plugging these values into the formula, we get:

C(12, 3) = 12! / (3!(12-3)!) = 12! / (3!9!) = (12 * 11 * 10) / (3 * 2 * 1) = 220

So, there are 220 ways to select 3 people from a group of 12.

a) Only Economists

To calculate the number of ways to select only economists, we need to choose 3 economists from the group of 4 economists. Using the combination formula, we get:

C(4, 3) = 4! / (3!(4-3)!) = 4! / (3!1!) = 4

So, there are 4 ways to select only economists.

b) Only Lawyers

To calculate the number of ways to select only lawyers, we need to choose 3 lawyers from the group of 8 lawyers. Using the combination formula, we get:

C(8, 3) = 8! / (3!(8-3)!) = 8! / (3!5!) = 56

So, there are 56 ways to select only lawyers.

c) Exactly Two Lawyers

To calculate the number of ways to select exactly two lawyers, we need to choose 2 lawyers from the group of 8 lawyers and 1 economist from the group of 4 economists. Using the combination formula, we get:

C(8, 2) * C(4, 1) = (8! / (2!(8-2)!)) * (4! / (1!(4-1)!)) = 28 * 4 = 112

So, there are 112 ways to select exactly two lawyers.

d) At Least One Economist

To calculate the number of ways to select at least one economist, we need to calculate the total number of ways to select 3 people minus the number of ways to select only lawyers. Using the values calculated above, we get:

Total number of ways to select 3 people = 220

Number of ways to select only lawyers = 56

Number of ways to select at least one economist = Total number of ways to select 3 people - Number of ways to select only lawyers = 220 - 56 = 164

So, there are 164 ways to select at least one economist.

Probability Calculation

To calculate the probability for each condition, we divide the number of ways to satisfy the condition by the total number of ways to select 3 people.

a) Probability of selecting only economists = Number of ways to select only economists / Total number of ways to select 3 people = 4 / 220 = 0.018

b) Probability of selecting only lawyers = Number of ways to select only lawyers / Total number of ways to select 3 people = 56 / 220 = 0.255

c) Probability of selecting exactly two lawyers = Number of ways to select exactly two lawyers / Total number of ways to select 3 people = 112 / 220 = 0.509

d) Probability of selecting at least one economist = Number of ways to select at least one economist / Total number of ways to select 3 people = 164 / 220 = 0.745

Answer

The probabilities for each condition are as follows: a) The probability of selecting only economists is approximately 0.018. b) The probability of selecting only lawyers is approximately 0.255. c) The probability of selecting exactly two lawyers is approximately 0.509. d) The probability of selecting at least one economist is approximately 0.745.

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