STAT 301 - Business Statistics
In-Class Activity - Sampling Distributions
(counts as Lecture Review #11)

Work with your partner(s) to answer the following questions. Group members should have their own individual writeups, which will be submitted (like regular review assignments) at the beginning of next class.

RECALL that an American-style roulette wheel contains 38 numbers — 1 through 36 (18 red, 18 black) plus 0 and 00 (which are green). We noted in class that many wagers are possible at the roulette table.

Question 1

One possible wager is to bet on "red" or "black." This wager pays "even money" — that is, if you bet $1 and win, you gain $1 net. (You get your dollar back, plus $1 more.) Of course, if you lose the bet your dollar is lost.

a) What is the expected value of a single $1 bet at red/black?

b) What is the expected variance of a single $1 bet at red/black?

(NOTE that we covered this in class, and so you should be able to find and follow the solution from your class notes.)

Question 2

Another possible wager is to bet on a column of twelve numbers. This wager pays "even money" — that is, if any of the twelve numbers comes up, you gain $2 net (your original dollar back, plus two more). And, of course, if another number comes up, you lose your dollar.

a) What is the expected value of a single $1 bet on a column of twelve numbers?

b) What is the expected variance of a single $1 bet on a column of twelve numbers?

c) How do these numbers compare with your results from Question 1? Does this make sense?

Question 3

Now suppose 10,000 separate wagers are placed on red or black, from 10,000 separate spins of the roulette wheel.

a) What will be the expected average winnings, per dollar bet? Why?

b) What will be the standard deviation of this quantity? Why?

c) In practice, you may be luckier (or less lucky) than this average. What is the probability that your average winnings, after 10,000 bets, is positive? That is, what’s the chance you make money after this extended time at the roulette table?

HINT: Remember the Central Limit Theorem: after this many separate random events, the sampling distribution of the sample average should be approximately normal. And remember the Law of Large Numbers: the standard deviation of the sampling distribution of the sample average will be σ/sqrt(n) .

Question 4

Now suppose 10,000 separate wagers are placed on columns of twelve numbers, from 10,000 separate spins of the roulette wheel.

a) What will be the expected average winnings, per dollar bet? Why?

b) What will be the standard deviation of this quantity? Why?

c) In practice, you may be luckier (or less lucky) than this average. What is the probability that your average winnings, after 10,000 bets, is positive? That is, what’s the chance you make money after this extended time at the roulette table?

d) How do these numbers compare with your results from Question 3? Does this make sense?

Let’s shift gears slightly now — to analyze the "martingale" gambling system we introduced in class.

Recall how the system works. You start off with an initial wager (here, we’ll bet $1) and play an even-money game (such as the red/black bet discussed in Question 1). If you win the bet, you quit and are $1 ahead. If you lose the bet, you double your wager and bet again. You keep doing this until you win … and you wind up $1 ahead. As we noted in class, you are "guaranteed" to win $1 … as long as you don’t run out of money first.

Suppose that you have $7, and plan to play the martingale system one time, with an initial wager of $1. Find the following quantities. (NOTE: you could do this on a calculator, but a spreadsheet might be an easier computational tool. NOTE ALSO: Keep lots of decimals on intermediate results, to avoid problems with roundoff error.)

a) The probability that you win $1, from playing the martingale system once, given your initial capital of $7. 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. (NOTE: This is not the expected value of a single bet. Rather, it is the expected value from one try at the system, which might involve multiple bets.)

c) Note that, playing the system, sometimes you wagered $1 to win $1 — but sometimes you had to wager $3 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.)