Probabilities

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Questions:

  1. (20 points) A standard deck has 52 playing cards. There are thirteen ranks, ranging from 2 up to 10, then Jack, Queen, King and Ace. Each rank has exactly one card each of four different suits: Spades ♠, Hearts ♥, Diamonds ♦, and Clubs ♣.

    a. Suppose you select two cards at random from the deck. What is the probability that you have a pair worse than a pair of Jacks? (The cards 2-10 are all worse than Jacks)

    1. The game of Pinochle (pronounced “pea-knuckle”) is played with a non-standard deck of cards. The deck only contains 9s, 10s, Jacks, Queens, Kings, and Aces. It contains two copies of each of these cards in each suit (so there are 2 copies of the 9♥, two copies of the K♣, and so on. An important card combination is the Pinochle, which consists of the Q♠ and the J♦. Suppose two cards are drawn at random from a Pinochle deck. What is the probability they form a Pinochle?

    2. (Bonus 10 points) The most popular card game in the world (at least for tournament competition) is Contract Bridge. In the game of Bridge, each of 4 players receives a hand of 13 cards from the standard 52 card deck. (Therefore the whole deck is dealt out to the players). Bridge is a very complicated game, but one important way in which players decide how to play their hand is by counting the number of “High Card Points”. Each ace is worth 4 points, each king 3, queen 2, and jack 1 point. Let X be the number of high card points a player has in their hand of 13 cards. Find E[X].

  2. (15 points) Using historical records, the manager of a construction site has determined that the number of minor injuries suffered by workers per day has the following distribution :

    Find the following probabilities:
    a) The number of injuries is in the interval [3,6]. b) Atleast1injuryoccurs.

c) Fewer than 4 injuries occur.

3. (20 points) You are a human resources professional. Your company is hiring for two different jobs. The first job is in sales, the second job is in finance. You have a list of candidates, who may be worth interviewing for one of the jobs, both of the jobs, or neither. The probability that a candidate is suitable for the sales job is .45, the probability that a candidate is suitable for the finance job is .35, and the probability of being suitable for both is .15. Answer the following questions:

a. What is the probability that a candidate is unhirable?
b. What is the probability that a candidate is worth interviewing for at least one job? c. What is the probability that a candidate is worth interviewing for the sales job, but

not the finance job?
d. Let X be the number of jobs for which a candidate is worth interviewing. Find E[X].

E

0

1

2

3

4

5

6

P(E)

.25

.25

.15

.10

.10

.08

.07

  1. (15 points) Actuaries are insurance professionals. Becoming an actuary requires passing a series of 9 exams. These exams are offered twice a year, and people traditionally take them while employed as junior actuaries. Therefore, actuarial jobs normally give workers paid study time. This can be a problem because study time can be used for activities other than studying, and studying is no guarantee one will pass a given exam. (It normally takes about 7 years to pass all 9 exams, which implies a failure rate of around 33% for people who actually pass all 9).

    Suppose 50% of all actuarial students study hard, and have an 80% chance of passing a given exam. The other 50% cram for the exam at the last minute, and have a 40% chance of passing the exam. You are the manager of some junior actuaries. One of them just took an exam and failed. What is the probability that they studied hard for the exam?

  2. (15 points) A marketing manager for a department store used a survey to determine how a sample of customers learned about the store’s products, as well as the gender of the customer. The joint probabilities are reported in the following table:

    1. a)  Given that a customer is female, what is the probability she learned about the store online?

    2. b)  Giventhatthecustomerlearnedaboutthestorevianewspaper,whatisthe probability they are male?

    3. c)  What is the probability a customer is female?

    4. d)  WhatistheprobabilityacustomerisbothmaleandlearnedaboutthestoreonTV?

    5. e)  Let E be the event {Online} and F be the event {Male}. Are E and F independent?

  3. (15 points) Suppose you are attending a party, which happens to be rather boring. You are talking with various people about their birthdays.

    a. Suppose there are 8 people attending the party. What is the probability that everyone was born in a different month? Assume that all partygoers will tell the truth about their birth month, and that people are equally likely to be born in any month.

b. Assume now that 10 people are attending the party. What is the probability that everyone was born on a different day of the month (For example, my birthday is January 18th. My day of the month would be the 18th.) For convenience, assume that all months have 30 days, and every day is equally likely to be someone’s birthday. 

 
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