Deck 7: Discrete Probability

Question 1

What is the probability that a randomly selected day of the year (366 days) is in May?

Answer 1

31/366.

Question 2

Prove or disprove:

Answer 2

False Choose one card at random from a deck of 52 cards. Let E = choose a diamond, F = choose a king.

Question 3

Let A be the set of all strings of decimal digits of length 5. For example 00312 and 19483 are two strings in
A.
You pick a string from A at random. What is the probability that
(a) the string begins with 7575.
(b) the string has no 4 in it.
In questions 13-16 you have 40 different books (20 math books, 15 history books, and 5 geography books).

Answer 3

Question 4

In questions suppose you have a class with 30 students 10 freshmen, 12 sophomores, and 8 juniors.
You pick two students at random, one at a time. What is the probability that both are freshmen?

Question 5

What is the probability that a randomly selected integer chosen from the first 100 positive integers is odd?

Question 6

In a certain lottery game, three distinct numbers between 10 and 25 (inclusive) are chosen as the winning
numbers. What is the probability that the winning numbers are all composite numbers.

Question 7

In questions suppose you have a class with 30 students 10 freshmen, 12 sophomores, and 8 juniors.
You pick two students at random, one at a time. What is the probability that the second student is a freshman,
given that the first is a freshman?

Question 8

What is the probability that a fair coin lands Heads 6 times in a row?

Question 9

What is the probability that a fair coin lands Heads 4 times out of 5 flips?

Question 10

You pick two books at random, one at a time. What is the probability that both books are history books?

Question 11

In questions suppose you have a class with 30 students 10 freshmen, 12 sophomores, and 8 juniors.
You pick one student at random. What is the probability that the student is not a junior?

Question 12

Find and correct the error in the solution to the following problem:
Problem: You flip two coins and want to find the probability that both coins show heads.

Question 13

You pick one book at random. What is the probability that the book is a history book?

Question 14

Suppose you pick two cards, one at a time, at random, from an ordinary deck of 52 cards. Find
(a) p(both cards are diamonds).
(b) p(the cards form a pair).

Question 15

You pick one book at random. What is the probability that the book is not a geography book?

Question 16

Three coins are tossed.
(a) List the elements in the sample space.
(b) Find the probability that exactly two heads show.

Question 17

What is the probability that a card chosen from an ordinary deck of 52 cards is an ace?

Question 18

What is the probability that the sum of the numbers on two dice is even when they are rolled?

Question 19

You pick two books at random, one at a time. What is the probability that the two books are from different
disciplines?

Question 20

Suppose you and a friend each choose at random an integer between 1 and 8, inclusive. For example, some
possibilities are (3, 7), (7, 3), (4, 4), (8, 1), where your number is written first and your friend's number second.
Find the following probabilities:

Question 21

In a certain lottery game you choose a set of six numbers out of 54 numbers. Find the probability that none

Question 22

A group of ten women and ten men are in a room. If five of the 20 are selected at random and put in a row
for a picture, what is the probability that the five are of the same sex?

Question 23

you flip a biased coin, where p(heads) = 3/4 and p(tails) = 1/4, ten times.
Find

Question 24

You pick a word at random from the set of all words of length six of letters of the alphabet with no repeated
letters. What is the probability that the word begins and ends with a vowel?

Question 25

of your numbers match the six winning numbers.
In questions you pick a bit string from the set of all bit strings of length ten.
What is the probability that the bit string has exactly two 1's, given that the string begins with a 1?

Question 26

of your numbers match the six winning numbers.
In questions you pick a bit string from the set of all bit strings of length ten.
What is the probability that the bit string has more 0's than 1's?

Question 27

an experiment consists of picking at random a bit string of length five. Consider the following events:
E1: the bit string chosen begins with 1;
E2: the bit string chosen ends with 1;
E3: the bit string chosen has exactly three 1's.

Question 28

an experiment consists of picking at random a bit string of length five. Consider the following events:
E1: the bit string chosen begins with 1;
E2: the bit string chosen ends with 1;
E3: the bit string chosen has exactly three 1's.

Question 29

an experiment consists of picking at random a bit string of length five. Consider the following events:
E1: the bit string chosen begins with 1;
E2: the bit string chosen ends with 1;
E3: the bit string chosen has exactly three 1's.

Question 30

A group of ten women and ten men are in a room. A committee of four is chosen at random. Find the
probability that the committee consists only of women?

Question 31

of your numbers match the six winning numbers.
In questions you pick a bit string from the set of all bit strings of length ten.
What is the probability that the bit string begins and ends with 0?

Question 32

of your numbers match the six winning numbers.
In questions you pick a bit string from the set of all bit strings of length ten.
What is the probability that the bit string begins with 111?

Question 33

an experiment consists of picking at random a bit string of length five. Consider the following events:
E1: the bit string chosen begins with 1;
E2: the bit string chosen ends with 1;
E3: the bit string chosen has exactly three 1's.
Determine whether E2 and E3 are independent.

Question 34

A red and a green die are rolled. What is the probability of getting a sum of six, given that the number on
the red die is even.

Question 35

A red and a green die are rolled. What is the probability of getting a sum of six, given that the number on
the green die is odd?

Question 36

of your numbers match the six winning numbers.
In questions you pick a bit string from the set of all bit strings of length ten.
What is the probability that the bit string has the sum of its digits equal to seven?

Question 37

an experiment consists of picking at random a bit string of length five. Consider the following events:
E1: the bit string chosen begins with 1;
E2: the bit string chosen ends with 1;
E3: the bit string chosen has exactly three 1's.

Question 38

You pick a word at random from the set of all words of length six of letters of the alphabet with no repeated
letters. What is the probability that the word has exactly one vowel?

Question 39

an experiment consists of picking at random a bit string of length five. Consider the following events:
E1: the bit string chosen begins with 1;
E2: the bit string chosen ends with 1;
E3: the bit string chosen has exactly three 1's.
Determine whether E1 and E2 are independent.

Question 40

of your numbers match the six winning numbers.
In questions you pick a bit string from the set of all bit strings of length ten.
What is the probability that the bit string has exactly two 1's?

Question 41

you flip a biased coin, where p(heads) = 3/4 and p(tails) = 1/4, ten times.
Find p(at least 7 heads).

Question 42

Urn 1 contains 2 blue tokens and 8 red tokens; urn 2 contains 12 blue tokens and 3 red tokens. You roll a die
to determine which urn to choose: if you roll a 1 or 2 you choose urn 1; if you roll a 3, 4, 5, or 6 you choose
urn 2. Once the urn is chosen, you draw out a token at random from that urn. Given that the token is blue,
what is the probability that the token came from urn 1?
In questions 45-47 a bowl has eight ping pong balls numbered 1,2,2,3,4,5,5,5. You pick a ball at random.

Question 43

you flip a biased coin, where p(heads) = 3/4 and p(tails) = 1/4, ten times.
Find

Question 44

Urn 1 contains 2 blue tokens and 8 red tokens; urn 2 contains 12 blue tokens and 3 red tokens. You pick an
urn at random and draw out a token at random from that urn. Given that the token is blue, what is the
probability that the token came from urn 1?

Question 45

You have two decks of 26 cards. Each card in each of the two decks has a different letter of the alphabet on
it. You pick at random one card from each of the two decks. A vowel is worth 3 points and a consonant is
worth 0 points. Let X = the sum of the values of the two cards picked. Find E(X), V (X), and the standard
deviation of X.

Question 46

A die has the numbers 1, 2, 2, 3, 3, 3 on its six sides. If the die is rolled, what is the expected value and variance
of the number showing?

Question 47

Find E(X), where X = the number on the ball you draw.

Question 48

Find p(the number on the ball drawn is >3 ).

Question 49

Find p(the number on the ball drawn is even).

Question 50

Each of 26 cards has a different letter of the alphabet on it. You pick one card at random. A vowel is worth 3
points and a consonant is worth 0 points. Let X = the value of the card picked. Find E(X), V (X), and the
standard deviation of X.

Question 51

A pair of dice, each with the numbers 1, 2, 2, 3, 3, 3 on its six sides are rolled.
(a) What is the expected value of the sum of the numbers showing?
(b) What is the expected value of the product of the numbers showing in part (a)?

Question 52

You flip a coin. If it lands heads, you lose 1 point. If it lands tails, you flip the coin again, and lose 1 point if
it lands heads and get 3 points if it lands tails. What is the expected value of the number of points you get
when you play this game.

Question 53

You have seven cards, numbered 3 through 9, and you pick one at random. If you pick a card with a prime
number, you get 1 point; if you pick a card with a composite number, you lose 1 point. Find the expected
value of the number of points you get.