Question

sixty five percent of people pass the state driver's exam on the first try. A group of 50 individuals who have have taken the driver's exam is randomly selected. Determine the expected value. What is the probability that exactly 25 people pass the test? more than 35?

Answer 1

Ans:

Use binomial distribution with n=50 and p=0.65

Expected value=np=50*0.65=32.5

P(exactly 25 pass test)=50C₂₅*0.65^25*(1-0.65)^25=0.0106

or use excel formula BINOMDIST(25,50,0.65,FALSE)=0.0106

P(more than 35)=1-P(less than equal to 35)=1-BINOMDIST(35,50,0.65,TRUE)=0.1878

Answer 2

Given:

• The probability of passing the driver's exam on the first try is 0.65

• The number of individuals in the selected group is 50.

To determine the expected value, we can use the formula:

Expected value = Probability of success x Total number of trials

Therefore, the expected value is:

Expected value = 0.65 x 50 = 32.5

The probability of exactly 25 people passing the test can be calculated using the binomial probability formula:

P(X = x) = (nCx) * p^x * (1 - p)^(n-x)

where:

• n = total number of trials = 50

• x = number of successes = 25

• p = probability of success = 0.65

• nCx = number of combinations of n things taken x at a time = n! / x!(n-x)!

Plugging in the values, we get:

P(X = 25) = (50C25) * 0.65^25 * (1 - 0.65)^(50-25) = 0.098

Therefore, the probability of exactly 25 people passing the test is 0.098 or approximately 9.8%.

To find the probability of more than 35 people passing the test, we can calculate the probability of 35 or fewer people passing the test and subtract it from 1. Using the cumulative distribution function of the binomial distribution, we get:

P(X <= 35) = Σ(i=0 to 35) (50Ci) * 0.65^i * (1-0.65)^(50-i) = 0.923

So, the probability of more than 35 people passing the test is:

P(X > 35) = 1 - P(X <= 35) = 1 - 0.923 = 0.077 or approximately 7.7%.