A youth group has 7 boys

and 5 girls. Experiment is select 5 people at random from the group. Let X be the number
of boys in the selected 5 people.
(a) Find the probability of exactly 1 boy, i.e., P([X = 1]).
(b) Find the probability of exactly 2 boy, i.e., P([X = 2]).
(c) Find the probability of at least one boy, i.e., P([X ≥ 1]).
(d) Find the expected value
 X
and the standard deviation
 X

prob(exactly one boy)

= C(7,1)xC(5,4)/C(12,5)
= 7(5)/792
= 35/792

prob(exactly 2 boys)
= C(7,2)xC(5,3)/792
= 21(10)/792
= 35/132

at least one boy
= 1 - prob(all girls)
= 1 - 1/792
=791/792

To solve this problem, we will use the concept of combinations and apply the probability mass function.

(a) Probability of exactly 1 boy, P(X = 1):

To determine the probability of exactly 1 boy, we need to consider that out of the 7 boys and 5 girls, we need to select 1 boy and 4 other people who can be either boys or girls.

The total number of ways to select 5 people from the group of 12 is given by the combination formula C(12, 5).

P(X = 1) = (Number of ways to select 1 boy and 4 other people) / (Total number of ways to select 5 people)

Number of ways to select 1 boy and 4 other people = C(7, 1) * C(5, 4)
Total number of ways to select 5 people = C(12, 5)

Therefore,
P(X = 1) = (C(7, 1) * C(5, 4)) / C(12, 5)

(b) Probability of exactly 2 boys, P(X = 2):

To determine the probability of exactly 2 boys, we need to consider that out of the 7 boys and 5 girls, we need to select 2 boys and 3 other people who can be either boys or girls.

P(X = 2) = (Number of ways to select 2 boys and 3 other people) / (Total number of ways to select 5 people)

Number of ways to select 2 boys and 3 other people = C(7, 2) * C(5, 3)
Total number of ways to select 5 people = C(12, 5)

Therefore,
P(X = 2) = (C(7, 2) * C(5, 3)) / C(12, 5)

(c) Probability of at least one boy, P(X ≥ 1):

To determine the probability of at least one boy, we can subtract the probability of selecting no boys from 1.

P(X ≥ 1) = 1 - P(X = 0)

To calculate P(X = 0), we need to consider that out of the 7 boys and 5 girls, we need to select 5 girls.

P(X = 0) = (Number of ways to select 5 girls) / (Total number of ways to select 5 people)

Number of ways to select 5 girls = C(5, 5)
Total number of ways to select 5 people = C(12, 5)

Therefore,
P(X ≥ 1) = 1 - (C(5, 5) / C(12, 5))

(d) Expected value (µX) and standard deviation (σX):

To find the expected value (µX), we multiply each possible value of X (0, 1, 2, ..., 5) by its corresponding probability and sum up the results.

µX = 0 * P(X = 0) + 1 * P(X = 1) + 2 * P(X = 2) + ... + 5 * P(X = 5)

To find the standard deviation (σX), we can use the formula:

σX = sqrt(E[(X - µ)^2])

Where E[ ] denotes the expectation.

Calculating these values requires computations based on combinations, which can be calculated using mathematical formulas or programming languages like Python with libraries such as math or scipy.