## STAT 13

(Sec. 1a-1c)

Introduction to Statistical Methods for the Life and Health Sciences

## Instructor: Ivo Dinov, Asst. Prof.

Departments of Statistics & Neurology
 http://www.stat.ucla.edu/~dinov/

Due Date:

# Friday, Nov. 16, 2001, turn in after lecture

See the HW submission rules. On the front page include the following header.

• (HW_5_1) The casino game of Tai Sai is played by betting on various possible combinations of results when three dice are rolled. One possibility is to place a bet on a single number. In this case, the more times this number appears on the three dice, the greater the winnings. Here's an example of how it works: Consider a gambler who wishes to bet on the number six. For each dollar bet, the player wins twelve times his/her stake if three sixes are rolled on the three dice, two times his/her stake if two sixes are rolled on the three dice or one times his/her stake if only one six is rolled on the three dice. If no sixes are rolled, then the stake is lost.
• Let X be the return from a single bet with a \$5 stake. X has the following probability function:
 x -\$5 \$5 \$10 \$60 pr(X=x) 0.579 0.347 0.069 0.005
Compute E[X], sd[X] and the probability of making a positive return from a single bet.

• A player will make a profit overall if their mean return from placing a series of bets is positve. Let X_bar be the mean return from 100 separate bets with a \$5 stake.
• What are the values of E[X_bar] and SD[X_bar]?
• What is the approximate distribution of X_bar? What theorem was needed to decide this?
• Calculate the probability of, on average, making a positive return from placing one hundred separate bets with \$5 stakes.
• Repeat the calculations above using 1,000, 5,000 and 10,000 bets.
• Create a scatter plot of probability that the mean return from playing Tai Sai is positive (on y-axis) versus number of bets (on x-axis). Comment on what the plot appears to show.

\Ivo D. Dinov, Ph.D., Departments of Statistics and Neurology, UCLA School of Medicine/