1. Suppose that 5 cards are drawn at random from a 52 card deck. Let E be the event that all 5 cards have the same suit, and let F be the event that the at least one card is a heart.
Find P(E), P(F), P(E \intersect F) and P(E|F).
2. Suppose that when the temperature is 90 degrees, alligators give birth to female alligators 60% of the time. Find the probability that when it is 90 degrees, an alligator that gives birth to 10 babies, has 6 female babies and 4 male babies.
3. Suppose that 2 dice are rolled. Let E be the event that the numbers on the two dice differ by exactly one, for example 3 and 4. Let F be the event that the sum of the two numbers is odd. Are E and F independent events? Explain your answer.
4. Draw a tree diagram and use it to find the probability that a random card from a 52 card deck is a face card, given that it is not a queen.
1. Suppose I offer you the following bet. Should you take it? We each put down 1 dollar. You pick any four distinct numbers from 1,2...,9. I pick 5 numbers, and if I pick at least 2 of your numbers, I win the money.
2. Suppose that a 10 sided die is rolled twice. Let E be the event that doubles are rolled, and let F be the event that a sum of 10 is rolled. Find P(E), P(E \union F), P(E \intersect F), P(E|F) and P(F|E).
3. Given an example of events E and F with positive probabilites such that P(E|F)=0.
4. Defective computers are made 5% of the time. If a company buys 30 computers, what is the probability that at most one computer is defective?
5. Suppose that 8 couples are facing each other in two lines. If one line is blindfolded and then randomly scrambled, what is the probability that exactly 6 couples remain matched?
1. Suppose that boys and girls are born each with probability 1/2. What is the probability that a family of 6 consists of 3 boys and 3 girls? Has at least 3 boys?
2. Suppose that a commitee of 4 is chosen from 8 students at random. What is the probability that Cosmo and Osmo are both on the committee?
3. A single six-sided die is rolled three times. Let E be the event that all three rolls are different. Let F be the event that no 6 appears. Find P(E), P(F), P(E \union F) and P(E|F).
4. Suppose that 4 cards are drawn at random from a 52 card deck. Let E be the event that either all 4 Aces are drawn, or all four queens are drawn. Let F be the event that no Kings are drawn. Find P(E). Explain without calculating whether P(E|F) will be larger than P(E) or less than P(E).
1. For the pool of students on page 126 of the course booklet, use the data there to decide if the event of being brown eyed and the event of being female are independent. Also, calculate the conditional probability that a randomly chosen student is female given the the student is brown eyed.
2. This problem concerns random arrangements of the letters in the word PRECEDE.
(a) What is the probability that the three E's are consecutive?
(b) Given that the third letter is R, what is the probability the the three E's are consecutive?
3. The numbers from 1 to 8 are placed on the eight faces of an octahedral die. The die is not fair. Rather the probabilities of the various faces are proportional to the the numbers on the faces.
(a) What is the probability that the number 7 will be rolled?
(b) If the die in part (a) is rolled twice, what is the probability that the sum of the two rolls equals 15?
4. Four cards are chosen at random (without replacement) from a standard deck of playing cards. What is the probability of getting three of a kind-that is three cards of one rank and one card of another rank? You need not simplify your answer, rather you may leave the choose symbol in your answer.
5. An experiment consists of flipping a fair coin four times. Use probabilities, not just intuition, to decide whether the event that the first flip is heads and the event that exactly two flips are heads are independent.