Streaks in sports

Discussion

1. Using a die, generate sequences of 40 0's and 1's of the following types:

   (a) The first outcome is determined by the toss of a coin and then each successive outcome has probability 2/3 of being the same as the previous outcome

   (b) Each outcome is determined by the toss of a coin

   (c) The first outcome is determined by the toss of a coin and then each successive outcome probability 2/3 of being different from the previous outcome.

   Exchange sequences with another group and see if you can figure out the types.

2. Find the probability that two people in your group have the same birthday.

3. Estimate the probability that two people in the class have the same birthday.

4. Is it reasonable to assume that a randomly chosen person's birthday is equally likely to fall on any of the 365 possible days of the year? If not, why not.

5. Read the following news article and see if you agree with the conclusions.

   Proprietary to the United Press International September1981 1-in-100 million chance

   Lottery officials say there is a 1-in-100 million chance the same 4-digit lottery numbers would be drawn in Massachusetts and New Hampshire on the same night.

   But that's just what happened Tuesday, when the number 8092 came up.

   ``There is a 1-in-10,000 chance of any 4-digit number being chosen at any given time,'' said Massachusetts Lottery Commission spokesman David Ellis.

   ``But the odds of it happening with two states at any one time are just fantastic,'' he said.

   Fifteen states currently have lotteries, Ellis said, of which 13 have numbers games similar to Massachusetts and New Hampshire.

   But, said Tom O'Heir, the director of the Lottery's computer services, ``the odds of the same number being drawn between New Hampshire and Massachusetts is 1-in-100 million.''

   For an exact match, the number paid $5,842 in Massachusetts and $4,500 in New Hampshire.

Journal assignment

1. Recall that in a football pool you have to pick the winner taking into account the point spread. For example, if New York is favored by 3.5 points over Washington and you bet on New York, you win the amount you bet if New York wins by more than 3 points and otherwise you lose the amount you bet. If you bet on Washington you win if Washington wins or loses by less than 4 points. In the ``teaser bet" you are asked to predict the winner in three games. For each game you get an extra 7 points on the point spread. This means that, if you bet on New York, you win if New York wins or loses by less than 4 points. If you bet on Washington, you win if Washington wins or loses by less than 11 points. In return for the more generous point spread, to win the ``teaser" bet you must be correct on all three of your bets.

   Hal Stern has shown that if you look at how football scores differ from the point spread, they have approximately a normal distribution with mean 0 and standard deviation 13.86. Using this, decide if the ``teaser" bet is a favorable, fair or unfavorable bet.