Deck 7: A: Discrete Probability

Question 1

you have 40 different books (20 math books, 15 history books, and 5 geography books).
You pick one book at random. What is the probability that the book is a history book?

Answer 1

15/40

Question 2

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

Answer 2

31/366

Question 3

In a certain lottery game, you choose a set of four different integers between 1 and 50, inclusive, and a fifth integer between 1 and 20, inclusive, which may be the same as one of the other four. (a) What is the probability you win the jackpot by matching all five numbers drawn? (b) What is the probability that you match three of the first four numbers, but not the fifth?

Answer 3

Question 4

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

Question 5

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 6

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

Question 7

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

Question 8

you have 40 different books (20 math books, 15 history books, and 5 geography books).
You pick two books at random, one at a time. What is the probability that both books are history books?

Question 9

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

Question 10

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

Question 11

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

Question 12

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 13

you have 40 different books (20 math books, 15 history books, and 5 geography books).
You pick one book at random. What is the probability that the book is not a geography book?

Question 14

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 15

Prove or disprove:

Question 16

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

Question 17

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.

Question 18

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. Solution: There are three possible outcomes: 2 heads, 2 tails, or 1 head and 1 tail. Since a "success" is one of these three outcomes,

Question 19

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 20

you have 40 different books (20 math books, 15 history books, and 5 geography books).
You pick two books at random, one at a time. What is the probability that the two books are from different disciplines?

Question 21

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 22

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

Question 23

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 24

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 25

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 26

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 27

Determine whether are independent.

Question 28

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

Question 29

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 30

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 31

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

Question 32

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 33

In 35-40 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 34

Determine whether E2 and E3 are independent.

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

Find .

Question 37

Find .

Question 38

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 39

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 40

Find .

Question 41

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

Question 42

a bowl has eight ping pong balls numbered 1, 2, 2, 3, 4, 5, 5, 5. You pick a ball at random.
Find p(the number on the ball drawn is ).

Question 43

a bowl has eight ping pong balls numbered 1, 2, 2, 3, 4, 5, 5, 5. You pick a ball at random.
Find E(X), where X = the number on the ball you draw.

Question 44

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 45

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 46

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.

Question 47

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 48

a bowl has eight ping pong balls numbered 1, 2, 2, 3, 4, 5, 5, 5. You pick a ball at random.
Find p(the number on the ball drawn is even).

Question 49

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 50

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?