Simple Probability Calculus

1. In craps (a game played with a pair of dice) you lose if you roll a 2 or an 11 on the first roll. What is the probability of losing on the first roll?
2. Two brothers skip school one morning and give the teacher the false excuse that they had a flat tire. The teacher separates them and asks them to each write down which tire was flat. What is the probability that they both chose the same tire (assuming that they both guess)?
3. What is the probability of winning the jackpot on the National Lottery? (To win the jackpot, one must match six different numbers between 1 and 49 in any order).
4. Suppose in a given week the expected payoff in the National Lottery is 8 million pounds (reduced slightly from the jackpot due to possibility of shared prizes). Suppose a person buys a ticket for one pound. Using probability as a model of belief, what can you say about that person's belief that they will win as compared to the actual odds of winning? (Assume that the person behaves rationally and plays the lottery solely to win. Are there other assumptions required to answer this question?)
5. Suppose it is the case that 19% of females smoke and 35% have blond hair. What is the probability that a randomly selected woman is both a smoker and blond? What is the probability that she is either?
6. Make a histogram of the data in section 5.3 of the probability notes. What is your estimate of the probability of the value 35?