Question

41​% of men consider themselves professional baseball fans. You randomly select 10 men and ask each if he considers himself a professional baseball fan. Find the probability that the number who consider themselves baseball fans is​ (a) exactly​ five, (b) at least​ six, and​ (c) less than four.

Answer 1

To find the probability that a certain number of men consider themselves professional baseball fans, we can use the binomial probability formula. We substitute the given values into the formula to find the probabilities of (a) exactly 5 men, (b) at least 6 men, and (c) less than 4 men considering themselves baseball fans.

Step-by-step explanation:

To find the probability of the number of men who consider themselves professional baseball fans, we can use the binomial probability formula. The probability of exactly k successes in n trials is given by:

P(k) = (nCk)(pk)(qn-k)

where nCk represents the number of combinations of k items from a set of n items, p is the probability of success, and q is the probability of failure.

(a) To find the probability of exactly five men considering themselves baseball fans, we substitute k = 5, n = 10, p = 0.41, and q = 1 - p into the formula:

P(5) = (10C5)(0.415)(0.5910-5)

(b) To find the probability of at least six men considering themselves baseball fans, we need to find the probabilities of exactly six, seven, eight, nine, and ten men considering themselves baseball fans, and sum them up using the formula:

P(at least 6) = P(6) + P(7) + P(8) + P(9) + P(10)

(c) To find the probability of less than four men considering themselves baseball fans, we need to find the probabilities of exactly zero, one, two, and three men considering themselves baseball fans, and sum them up using the formula:

P(less than 4) = P(0) + P(1) + P(2) + P(3)

Answer 2

a)
`P(X=5)=0.208`

b)
`P(X\geq 6)=0.1854`

c)
`P(X<4)=0.3566`

Step-by-step explanation:

Given : 41​% of men consider themselves professional baseball fans. You randomly select 10 men and ask each if he considers himself a professional baseball fan.

To Find : The probability that the number who consider themselves baseball fans is​ (a) exactly​ five, (b) at least​ six, and​ (c) less than four.

Solution :

Applying binomial theorem,

`P(X=r)=^nC_r* p^r* (1-p)^(n-r)`

The success p= 41%=0.41

The failure (1-p)=1-041=0.59

Number of selection n=10

a) The probability that the number who consider themselves baseball fans is​ exactly​ five,

i.e. X=5

`P(X=5)=^(10)C_5* (0.41)^5* (0.59)^(10-5)`

`P(X=5)=(10!)/(5!5!)* (0.41)^5* (0.59)^(5)`

`P(X=5)=252* 0.0115* 0.071`

`P(X=5)=0.208`

b) The probability that the number who consider themselves baseball fans is​ at least​ six,

i.e.
`X\geq 6`

`P(X\geq 6)=1-P(X<6)`

`P(X\geq 6)=1-[P(X=0)+P(X=1)+P(X=2)+P(X=3)+P(X=4)+P(X=5)]`

`P(X\geq 6)=1-[^(10)C_0* (0.41)^0* (0.59)^(10-0)+^(10)C_1* (0.41)^1* (0.59)^(10-1)+^(10)C_2* (0.41)^2* (0.59)^(10-2)+^(10)C_3* (0.41)^3* (0.59)^(10-3)+^(10)C_4* (0.41)^4* (0.59)^(10-4)+^(10)C_5* (0.41)^5* (0.59)^(10-5)]`

`P(X\geq 6)=1-[0.0051+0.0355+0.111+0.205+0.250+0.208]`

`P(X\geq 6)=1-[0.8146]`

`P(X\geq 6)=0.1854`

c) The probability that the number who consider themselves baseball fans is​ less than four,

i.e.
`X<4`

`P(X<4)=P(=0)+P(X=1)+P(X=2)+P(X=3)`

`P(X<4)=^(10)C_0* (0.41)^0* (0.59)^(10-0)+^(10)C_1* (0.41)^1* (0.59)^(10-1)+^(10)C_2* (0.41)^2* (0.59)^(10-2)+^(10)C_3* (0.41)^3* (0.59)^(10-3)`

`P(X<4)=0.0051+0.0355+0.111+0.205`

`P(X<4)=0.3566`