Exam 7: Discrete Probability

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).

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?

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?

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

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.

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

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?

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. 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?

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?

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

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?

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.

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. - $\text { Find } p \left( E _ { 3 } \mid E _ { 2 } \right)$

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.

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.

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

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

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.

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?