DS280 - Intro to Statistics

Homework #9 - Random Variation

This assignment is due at the beginning of class on Friday, November 6. (This is later than the original course schedule.) It is worth 40 points – the equivalent of two quizzes. The two problems are equally weighted.

As always, while you are encouraged to work together with other students, your final writeup must be your own, in your own words. (Rule of thumb: if I can tell at a glance whom you worked with, you have gone beyond the bounds of acceptable collaboration.) This assignment need not be typed, but may be neatly handwritten. Please put the two problems on separate pages, to make the grading easier.

Question 1:

In class we discovered the “martingale” system of betting at roulette (or any other game that offers “even money” odds – a $1 bet wins $1 net). Recall that the system involved doubling the bet until you won, and then stopping. You were “guaranteed” to win – that is, unless you ran out of money in an extended string of losses.

Suppose that you have $100, and plan to play the martingale system one time, with an initial wager of $1. You decide to bet on “red.” (Recall that the roulette wheel has 18 “red” numbers out of the 38.) Find:

a) The probability that you win $1, from playing the martingale system once, given your initial capital of $100. That is, find the probability that you win a roulette bet before you run out of money.

b) The expected value of your net winnings, from one play at the martingale system.

c) Note that, playing the system, sometimes you wagered $1 to win $1 – but sometimes you had to wager $3 or $7 or more to win $1. Find the expected value of the amount wagered.

d) From your results in (b) and (c), find the expected value of the winnings per dollar bet. (Your answer should look familiar, from our in-class roulette examples).

NOTE that the calculations here are potentially subject to severe round-off error. I suggest that you retain a lot of decimal places on your intermediate answers, so your final answer to Part D is not thrown off.

Question 2:

“Skunk” is a recreational (rather than a casino) dice game. The player rolls a pair of dice, and scores points depending on the outcome of the rolls. Specifically:

1) If neither of the dice shows a 1, then the player gets points equivalent to the total of the two dice. Thus, rolling a 3 and a 5 gives eight points.

2) Assuming no 1 was thrown, the player can then choose to stop (in which case s/he gets the points that have been accumulated) or to continue rolling to try to accumulate more points. Thus, if the player rolls a 3 and a 5 the first time, decides to continue, and then rolls a 2 and a 3, this gives a total of thirteen points, with the option of trying to score even more.

3) If either of the dice is a 1 the player loses her/his turn, and all of the points accumulated in the turn. Thus, if the first roll were a 3 and 5 but the second roll was a 1 and a 6, the player would have to stop rolling and would get NO points for the turn.

4) If both dice are 1’s, then the player immediately loses her/his turn, as well as all points accumulated to this point in the game (not just the points in the current round).

a) Suppose it is the first turn of the game. What is the expected value of the number of points that you score on your very first roll? What is the expected variance of this quantity?

b) Suppose it is the first turn of the game. You rolled 3 and 5 on your first roll, and have decided roll one more time. What is the expected value of the number of points you score on this one additional role? (Remember, if either of the dice shows a 1, you will lose the eight points you have from your first roll.) What is the expected variance of this quantity?

c) Now suppose that it is the second turn of the game, and that you scored twelve points on your first turn. What is the expected value of the number of points you score on your very first roll? (Remember, a double-1 will lose you the twelve points you had at the start of your turn.) What is the expected variance of this quantity?