Question 1

A university in your region estimates that graduating average of high school students who apply for admission to a particular university program can be described by a Normal model with a mean of 82% and a standard deviation of 5%.The staff in admissions opens the applications at random looking for 10 applicants with averages above 85%.How many applications do you think the staff will need to open?
A)About 36 applications
B)About 21 applications
C)About 10 applications
D)About 28 applications
E)About 14 applications

Accepted Answer

To find the number of applications needed, first calculate the probability of a student having an average above 85%. Using the Z-score formula: Z = (X - μ) / σ, where X is 85, μ is 82, and σ is 5, we get Z = (85 - 82) / 5 = 0.6. Looking up 0.6 on the Z-table gives a probability of about 0.7257 to the left, so the probability to the right (above 85%) is 1 - 0.7257 = 0.2743. To find one student with an average above 85%, we expect 1 / 0.2743 ≈ 3.64 applications. For 10 students, multiply by 10, giving approximately 36.4 applications, which rounds to about 36 applications.

Question 2

Five men and four women are waiting to be interviewed for jobs.If they are all selected in random order,find the probability that the last person interviewed will be a man.Your simulation should use at least 10 runs.
A)1
B) $\frac { 5 } { 9 }$
C) $\frac { 4 } { 9 }$
D)0
E) $\frac { 1 } { 9 }$

Accepted Answer

The probability that the last person interviewed is a man is $\frac{5}{9}$ because there are 5 men and 4 women, making a total of 9 people. The selection of the last person is independent of the order in which the others are selected, so each person has an equal chance of being last.

Question 3

Eight men and three women are waiting to be interviewed for jobs.If they are all selected in random order,find the probability that the first person interviewed will be a woman.Your simulation should use at least 10 runs.
A) $\frac { 1 } { 11 }$
B)1
C) $\frac { 3 } { 11 }$
D)0
E) $\frac { 5 } { 11 }$

Accepted Answer

We can use the formula for probability:

P(a woman is first) = (# of ways a woman can be first) / (total # of ways)

Since there are 3 women and 11 people total, there are 3 ways to choose a woman to be first.
Then, the total number of ways to order all 11 people is 11! (11 factorial).

So, the probability is:

P(a woman is first) = 3 / 11!

Using Python to simulate at least 10 runs of shuffling a list of 8 "M" for men and 3 "F" for women:

import random

n_trials = 10
successes = 0

for i in range(n_trials):
    people = ["M", "M", "M", "M", "M", "M", "M", "M", "F", "F", "F"]
    random.shuffle(people)
    if people[0] == "F":
        successes += 1
        
print("Simulation Probability:", successes/n_trials)

The simulation probability should be close to the calculated probability of 3/11!, which is approximately 0.00288.

Question 4

The lengths of long distance phone calls placed on a cell phone can be described by a Normal model with a mean of 7 minutes and a standard deviation of 2 minutes.If one call is randomly selected from wireless phone company records,what is the probability that it will have lasted more than 11 minutes?
A)0.0977
B)0.23
C)0.977
D)0.023
E)0.98

Accepted Answer

We need to find P(X > 11), where X represents the length of a long distance phone call. We can standardize the variable by using z-score:
z = (11 - 7) / 2 = 2
Now we can look up the probability of z being greater than 2 in the standard Normal table, which gives us 0.0228. Therefore, P(X > 11) = 0.0228 or 0.023 when rounded to three decimal places. So, the correct choice is D.

Question 5

Six men and three women are waiting to be interviewed for jobs.If they are all selected in random order,find the probability that all the women will be interviewed first.Your simulation should use at least 10 runs.
A) $\frac { 1 } { 56 }$
B) $\frac { 1 } { 84 }$
C) $\frac { 1 } { 14 }$
D) $\frac { 1 } { 28 }$
E) $\frac { 2 } { 3 }$

Accepted Answer

The probability that all women will be interviewed first is calculated by considering the total ways to arrange the women (which is always 1 way since all must go first) over the total ways to arrange all 9 people. The total ways to arrange 9 people is 9!, and since the order of the 3 women doesn't matter when they are all interviewed first, we divide by the ways to arrange the 6 men, which is 6!. Therefore, the probability is 1/(9!/(3!6!)) = 1/84.

Question 6

What is the probability that Jennifer will have two sons out of her three children? Assume that boy and girl babies are equally likely.Disregard the likelihood that there are more male births and that there is a higher infant mortality rate for boys.Your simulation should use at least 10 runs.
A) $\frac { 1 } { 4 }$
B) $\frac { 3 } { 8 }$
C) $\frac { 1 } { 2 }$
D) $\frac { 5 } { 8 }$
E) $\frac { 1 } { 8 }$

Accepted Answer

The probability of having two sons out of three children, assuming equal likelihood for boy and girl births, can be calculated by considering all possible outcomes. There are $2^3 = 8$ possible outcomes (BBB, BBG, BGB, GBB, BGG, GBG, GGB, GGG), and 3 of these outcomes contain exactly two boys (BBG, BGB, GBB). Therefore, the probability is $\frac{3}{8}$.

Quiz 9: Understanding Randomness

Five men and four women are waiting to be interviewed for jobs.If they are all selected in random order,find the probability that the last person interviewed will be a man.Your simulation should use at least 10 runs.

Eight men and three women are waiting to be interviewed for jobs.If they are all selected in random order,find the probability that the first person interviewed will be a woman.Your simulation should use at least 10 runs.

Six men and three women are waiting to be interviewed for jobs.If they are all selected in random order,find the probability that all the women will be interviewed first.Your simulation should use at least 10 runs.

What is the probability that Jennifer will have two sons out of her three children? Assume that boy and girl babies are equally likely.Disregard the likelihood that there are more male births and that there is a higher infant mortality rate for boys.Your simulation should use at least 10 runs.