Practice Problems - Probabilities, Counting, and Equally Likely Outcomes

Weighted Probability - from 2.1

A random experiment has sample space = {O1, O2, O3, O4, O5}. Outcome 1 is twice as likely as outcome 2. Outcome 2 is as likely as outcome 3 and outcome 4. Outcome 5 is three times as likely as outcome 2. Find Pr(O4).

Weighted Probability Easy Method - from 2.1

A random experiment has sample space = {O1, O2, O3, O4, O5}. Outcome 1 is twice as likely as outcome 2. Outcome 2 is as likely as outcome 3 and outcome 4. Outcome 5 is three times as likely as outcome 2. Find Pr(O4).

Probabilities of Weighted Outcomes – Unfair Die - from 2.1

An unfair (six sided) die is such that the outcomes 1,2,3,4 are equally likely, 5 is half as likely as 2, and 6 is four times as likely as 4. Find Pr(1).

Permutation Arrangement – Employee Trip - from 2.2

You have 10 employees. How many ways can you select 2 of them for a trip to New York and one of them for a trip to Chicago? No employee gets to go to both cities.

Permutation Arrangements With Multiple Scenarios – Three Letters - from 2.2

A three-letter arrangement is made from the set {A, B, C, D, E}. How many possible arrangements are there, if each letter can be used at most twice?

Combination and Permutation Mixed Arrangements – Birds - from 2.2

3 crows, 3 jays, and 1 starling sit on a section of a telephone wire. In how many ways can they be arranged so that all the birds of the same type are sitting together? (assuming all crows and jays are distinct from each other)

Permutation Arrangements – Rearranging Letters of a Word - from 2.2

How many ways can the letters from the word “tattoo” be arranged into?

Probabilities of Combination Arrangements – King’s Island - from 2.4

There are 8 kids all of different ages. 4 are given King’s Island tickets. What is the probability that the 4 given the tickets were the youngest 4?

Probabilities of Combination Arrangements – Incomplete Deck - from 2.4

An incomplete deck of cards contains 4 spades, 7 hearts, 5 diamonds, and 5 clubs. Two cards are drawn at random without replacement. What is the probability that one is red and one is black? (Recall that hearts and diamonds are red and spades and clubs are black)

Probabilities of Combination Arrangements With Multiple Scenarios – Defective Parts - from 2.4

To assure product quality, a company checks a random 4 items they produce at the end of the day. They produce 10 items on a given day. If three of the items were defective, what is the probability that at least one defective item is detected?

Probabilities of Combination Arrangements – Delegation of Five - from 2.4

A Finite math class consists of 41 freshmen and 52 sophomores. The class decides that no enough homework is assigned, and a delegation of 5 is chosen at random to discuss the matter with the instructor of the class. Find the probability that no freshmen are included in this delegation.

Probabilities of Combination Arrangements – Socks in Drawer - from 2.4

Steve has 4 white socks and 6 black socks in a drawer, He selects 2 socks at random and puts them on. What is the probability that he wears socks of the same color?

Probabilities of Permutation Arrangements – Cast Auditioning - from 2.4

A cast has 2 female roles and 2 male roles in it. 4 males are auditioning, including John, and 3 females are auditioning, including Alice. What’s the probability exactly one of John and Alice are selected?

Probabilities of Permutation Arrangements – 4 Lane Race - from 2.4

Four runners enter two races, both to be run on a 4-lane track. The runners are assigned a lane for each race. Assuming the lane assignments are made at random, what is the probability each runner has the same lane assignment for both races?

Probabilities of Combination Arrangements – Standard Deck - from 2.4

A hand of 3 cards is drawn from a standard deck. How many ways can this be done when exactly 2 face cards are drawn?

Probabilities of Permutation Arrangements With Multiple Scenarios – Square Table - from 2.4

Alice, Betty, Cindy, and Diana are seated at a square table, with a seat on each side of the square. What’s the probability that Alice is seated right across from Betty?