Question 1

A company realizes that 5% of its pens are defective.In a package of 30 pens,is it likely that more than 6 are defective? Assume that pens in a package are independent of each other.

Accepted Answer

We can use the binomial distribution to solve this problem. Let X be the number of defective pens in a package of 30 pens. Then X follows a Binomial(30, 0.05) distribution.
P(X > 6) = 1 - P(X ≤ 6)
= 1 - (P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6))
= 1 - 0.9638
= 0.0362
Since P(X > 6) = 0.0362 is a low probability, it is likely that more than 6 pens are defective in a package of 30 pens. Therefore, the best choice is A.

Question 2

Suppose that in a certain population 8% of people are color blind.A researcher selects people at random from this population.What's the probability that the first color blind person will be found among the first 6 people checked?

Accepted Answer

The probability of finding a color blind person in the first check is 0.08. The probability of not finding a color blind person in the first check is 0.92. The probability of not finding a color blind person in the first 6 checks is 0.92^6 = 0.6064. The probability of finding a color blind person in the first 6 checks is 1 - 0.6064 = 0.3936.

Question 3

A basketball player has made 65% of his foul shots during the season.Assuming the shots are independent,find the probability that in tonight's game he makes his first basket on one of his first 4 shots.

Accepted Answer

D

Question 4

We draw a card from a deck 40 times to find the distribution of the suits.After each draw the card is replaced.

Accepted Answer

The answer of We draw a card from a deck...

Question 5

Suppose that in a certain population 45% of people have type O blood.A researcher selects people at random from this population.What is the probability that there is a person with type O blood among the first 6 people checked?

Accepted Answer

The probability that none of the first 6 people have type O blood is (0.55)^6. Therefore, the probability that at least one person has type O blood is 1 - (0.55)^6 = 0.9723.

Question 6

Suppose that 14% of people are left handed.If 8 people are picked at random,what's the probability that the first lefty is the 8th person chosen?

Accepted Answer

The probability that the first left-handed person is the 8th person chosen is calculated by having the first 7 people be right-handed (0.86 probability each) and the 8th person be left-handed (0.14 probability). This is (0.86^7) * 0.14 = 0.0487.

Question 7

A tennis player makes a successful first serve 90% of the time.Assume that each serve is independent of the others.If she serves 10 times,what is the probability that she misses for the first time on her second or fourth serve?

Accepted Answer

The probability of missing the first serve is 10%. The probability of missing the second serve is also 10%. The probability of missing the fourth serve is also 10%. The probability of missing the first time on the second or fourth serve is the sum of the probabilities of missing on the second and fourth serves, which is 10% + 10% = 20%.

Question 8

Suppose that in a certain population 46% of people have type O blood.A researcher selects people at random from this population.What's the probability they won't find a person with type O blood among the first 4 people checked?

Accepted Answer

The answer of Suppose that in a certain population 46%...

Question 9

An archer is able to hit the bull's-eye 71% of the time.If she shoots 10 arrows,what is the probability that her first bull's-eye comes on the 5th arrow? Assume each shot is independent of the others.

Accepted Answer

The probability that the bull's-eye comes on the 5th arrow is:
(0.29)^4 x (0.71) = 0.00502 
The (0.29)^4 is the probability of not hitting the bull's-eye for the first 4 shots (since she hits the bull's-eye 71% of the time, the probability of not hitting it is 1 - 0.71 = 0.29) and the (0.71) is the probability of hitting it on the 5th shot. Therefore, the best choice is B.

Question 10

You roll a die 7 times and need to get at least three 2's to win the game.

Accepted Answer

The probability of getting a 2 on a die roll is independent of previous rolls, and each roll has a fixed probability of 1/6 for landing on any given face. Therefore, the chance of getting a 2 does not change with each roll, making the rolls independent events.

Question 11

Suppose a computer chip manufacturer rejects 3% of the chips produced because they fail presale testing.What is the probability that the seventh chip you test is the first bad one you find?

Accepted Answer

The probability of finding a bad chip on the seventh test is 0.03 * 0.97^6 = 0.0250.

Question 12

A pool of possible jurors consists of 15 men and 18 women.A jury of 12 is picked at random from this group.What is the probability that the jury contains all women?

Accepted Answer

The probability of selecting a woman changes with each selection because the pool of potential jurors decreases in size, affecting the composition of men and women available for subsequent selections.

Question 13

We record the blood types (O,A,B,or AB)found in a group of 100 people.Assume that the people are unrelated to each other.

Accepted Answer

The answer of We record the blood types (O,A,B,or AB)found...

Question 14

We draw a card from a deck 6 times (replacing the card after each draw)and get 3 kings.How likely is this?

Accepted Answer

The answer of We draw a card from a deck...

Question 15

A study found that 56% of people working for large companies are dissatisfied with their job.Your manager conducts a survey at your company and finds that 225 of the 440 employees are dissatisfied with their job.

Accepted Answer

The fact that the employees are working for the same company means they are not independent of each other. Their job satisfaction may be influenced by factors specific to that company such as management, company culture, and work environment. Therefore, the assumption of independence necessary for using probability calculations is not valid in this case.

Question 16

We deal 6 cards from a deck and get 3 hearts.How likely is this?

Accepted Answer

The answer of We deal 6 cards from a deck...

Question 17

A basketball player has made 78% of his foul shots during the season.Assuming the shots are independent,find the probability that in tonight's game he misses for the first time on his 5th attempt?

Accepted Answer

Since the shots are independent, the probability of missing on the 5th attempt is the probability of making the first 4 shots and then missing the 5th.

The probability of making a shot is 0.78, so the probability of missing on any given attempt is 1 - 0.78 = 0.22.

Therefore, the probability of making the first 4 shots is (0.78)^4, and the probability of missing the 5th shot is 0.22.

The probability of making the first 4 shots and then missing the 5th is the product of these two probabilities:

(0.78)^4 * 0.22 = 0.0814

Therefore, the answer is B, 0.0814.

Question 18

Suppose a computer chip manufacturer rejects 4% of the chips produced because they fail presale testing.What's the probability you find a bad chip within the first 6 you examine?

Accepted Answer

The probability of finding a bad chip within the first 6 can be calculated using the binomial distribution formula, where n=6 (number of trials), p=0.04 (probability of success, i.e., finding a bad chip), and x=1 (number of successes, i.e. finding one bad chip). P(X=1) can be calculated as follows:

P(X=1) = nCx * p^x * (1-p)^(n-x)
= 6C1 * 0.04^1 * (1-0.04)^(6-1)
= 6 * 0.04 * 0.961^5
= 0.0217 (rounded to 4 decimal places)

Therefore, the probability of finding a bad chip within the first 6 is 0.0217, which is answer choice C).

Question 19

How likely is it that in a group of 400 people,the majority have blue eyes,given that 22% of the population have blue eyes? Assume that the people are unrelated to each other.

Accepted Answer

The answer of How likely is it that in a...

Question 20

We wish to predict the outcome of a local mayoral election,and poll 500 of the 4000 likely voters to see how many favor the incumbent.

Accepted Answer

The answer of We wish to predict the outcome of...

Quiz 16: Probability Models