Deck 5: Apecial Discrete Distributions

Question 1

Suppose that a certain car manufacturer makes 10 cars on some day; 4 being red and 6 being blue. A car dealer picks 5 cars at random from the 10 and purchases them. What is the probability that she gets at least 2 red cars?

Answer 1

$1 - \left( \begin{array} { l } 5 \\0\end{array} \right) ( .4 ) ^ { 0 } ( .6 ) ^ { 5 } - \left( \begin{array} { l } 5 \\1\end{array} \right) ( .4 ) ^ { 1 } ( .6 ) ^ { 4 } = .66$

Question 2

John claims that he is mildly psychic and can guess the color of a card withdrawn from an ordinary deck of cards correctly 70% of the time. His friend selects cards at random from the deck (replacing them and shuffling each time). He selects 12 cards and John guesses 9 correctly.
(a) Find the probability that John guesses exactly 9 correctly if his claim of being psychic is true.
(b) Find the probability that he gets exactly 9 correct if he is just guessing.

Answer 2

(a) $\left( \begin{array} { c } 12 \\9\end{array} \right) ( .7 ) ^ { 9 } ( .3 ) ^ { 3 }$
(b) $\left( \begin{array} { c } 12 \\9\end{array} \right) ( .5 ) ^ { 9 } ( .5 ) ^ { 3 }$

Question 3

People are in line for a free T-shirt. The company knows that 30% of people select a red shirt, 45% select a grey shirt, and 25% select a black shirt. Of the next 5 people who choose a shirt, what is the expected number of people who select a red shirt?

Answer 3

Expected Value is 5(.3)=1.5.

Question 4

You select cards randomly until you get the queen of spades, replacing the card and shuffling each time. What is the probability that it takes exactly 8 draws?

Question 5

Assume that busses arrive at a stop randomly and at a Poisson rate of 4 per hour. If they begin arriving at 9am, what is the probability that 5 busses arrive by 10:30 am?

Question 6

A batter gets a hit in 35% of her at-bats, independently of other at-bats. If X is the number of at-bats she must attempt to get 1 hit, what is the variance of X and E(X²)?

Question 7

Your friends went apple picking and gave you 10 apples from their stash. If their stash contained 50 good apples and 12 rotten apples, what is the probability that your share contained at least 2 rotten apples?

Question 8

Suppose that a certain book club has 12 members. Five of the members like romance and seven like mystery. If you select 4 members at random, what is the expected value of the members in the sample who like romance?

Question 9

In a large city, a radio station gets calls all day at a Poisson rate of 5 per hour. If they are giving out 20 tickets to the first 20 callers, what is the probability that they give away all the tickets in exactly 3 hours?

Question 10

In a certain community, 80% of residents hand out Halloween candies and 20% do not. If you visit 12 houses randomly, what is the probability that you get candy from at least 9 of them?

Question 11

Suppose that a low end dry cleaner has a 5% chance of losing any particular garment, independently of the other garments. You drop off 12 garments at this dry cleaner. What is the probability that you get at least 10 of them back?

Question 12

For not forfeiting a match, a university intramural team must have 3 female competitors and 3 male competitors. A certain team has 12 female members and 9 male members. If the female members show up for a match independently of one another with probability .25, what is the probability that at least 3 females arrive to the match.

Question 13

You walk around a large city asking people at random whether they are born in the same month as you were, but you stop doing this if you ask 25 people without anyone saying yes. Let X be the number of people you ask before someone says yes. (Assume that birthdays are equally distributed among the months.) Find P(X=5) and P(X=25).

Question 14

On a random day, of the dogs that are dropped off at a kennel, 45% are male, and of the male dogs, 33% are working breeds. Let X be the number of the first 8 dogs arriving at the kennel on a certain day and are male and a working breed. Find the probability mass function of X, its expected value and its variance.

Question 15

In a small New England town, a non-lethal virus comes through, and 1 in 100 people who contract the virus show symptoms. Find the probability that the third person to show symptoms is within the first 400 people.