1)There was an old TV game show called "Let's Make a Deal" where the contestants would dress in funny costumes and try to win prizes from the host Monty Hall.You are a contestant on this show and Monty says you will win the prize behind whichever one of three closed doors you pick.He tells you that two of the doors have a goat hidden behind them and the third door hides a new car.Obviously, you would rather win the car.You pick one of the doors and Monty says:"Before I show you what is behind the door you picked, I want to show you what's behind one of the doors you didn't pick."He opens one of the other doors and reveals a goat.Now he says:"Before I show you what you won, I will give you a chance to change your choice and switch to the other closed door, if you want."Should you take Monty up on his offer and switch?Why, or why not?Use Bayes' Theorem to provide your answer

2)You meet your new neighbor for the first time on the street.She is holding hands with a boy, whom she introduces to you as her son.She also mentions that she has one other child, who is currently at home.What is the probability that the other child is a girl?(You can assume that the probability of a boy or a girl being born is exactly .50-.50, even though that is not actually precisely true.)

3)Ricardo's sock drawer contains four black socks, eight blue socks and ten green socks. The individual socks are mixed up randomly in the drawer. Ricardo has to get up early to get ready for class and he does not want to wake his roommate by turning on the light, so he picks two socks at random from the drawer and wears them to class.What is the probability that Ricardo will go to class wearing matching socks?

4)You are in Las Vegas and need $35 to buy a great steak dinner, but you only have $20.You decide to play roulette to win the money you need for your dinner.In the simplest bet in roulette, you bet $1 on one of 38 numbers and if your number comes up on the wheel you win $35.If any other number comes up, you lose.

a)If you bet $1 on each of 20 different numbers on one single spin of the wheel, what is the probability that you will have $35 after your bet?

b)If you bet $1 on each of 5 different numbers on four different spins of the wheel, what is the probability that you will have at least $35 when you are done?

c)If you bet $1 on a single number for 20 spins of the wheel, what is the probability that you will have at least $35 when you are done?

d)Which is the best strategy to maximize your probability of getting that steak dinner?

5)In the Mega Millions lottery, a player buys a $1 ticket and picks five numbers from the numbers 1 to 70.The lottery has a bin with seventy white balls, each with a number from 1 to 70.The lottery picks five balls from the bin with the white balls.If the five numbers the player picked match the numbers drawn on the five white balls then the player wins $1,000,000 (in most states, but not including California, where the rules are different). In addition, the player picks one Mega number from the numbers 1 to 25.The lottery has a bin with twenty-five gold balls, each with a number from 1 to 25.The lottery picks one gold ball from the bin.If the player matches all five numbers on the white balls, plus the number on the gold ball, the player wins the Mega Millions jackpot (which is a minimum of $40 million). Assume you buy your ticket outside of California. Calculate the exact probability of winning exactly $1,000,000 (no more and no less) in the Mega Millions lottery if you buy one ticket.Include 11 decimal points in your answer.Hint:Use the Multiplication Law for Independent Events.

6)There are 36 students enrolled in this class.What is the probability that at least two of the students have the same birthday?(Ignore the possibility that someone might have been born on Feb. 29.)