1. Copyright © 2016, 2013, and 2010, Pearson Education, Inc.9
Chapter
Probability
Copyright © 2016, 2013, and 2010, Pearson Education, Inc.

2. 9-3 Simulations and Applications in Probability
Students will be able to understand and explain:
Simulations in probability;
Odds; and
Expected value.

3. Using Simulations in Probability
A simulation is a technique used to act out a problem by conducting experiments whose outcomes are analogous to the original problem.
Using simulations, students typically estimate a probability using many trials rather than determine probabilities theoretically.

4. Random Digits Table
The digits in this table form a list of digits selected at random, often by a computer or a calculator.

5. Random Digits Table
To simulate a coin toss, pick a number at random to start, and then read across the table, letting an even digit represent “heads” and an odd digit represent “tails.”

6. Example
A baseball player has a batting average of 0.400; that is, his probability of getting a hit on any particular time at bat is Estimate the probability that he will get at least one hit in his next three times at bat.

7. Example (continued)
Use a random-digit table. A hit is represented by the digits 0, 1, 2, and 3.
A 0, 1, 2, or 3 appears in 42 out of the 50 trials. An
estimate for the probability is

8. Example
Suppose Corey makes enough batter for exactly 100 chocolate chip cookies and mixes 100 chocolate chips into the batter. If the chips are distributed at random and Johnny chooses a cookie at random from the 100 cookies, estimate the probability that it will contain exactly one chocolate chip. (Call this event E.)

9. Example (continued)
Construct a 10 x 10 grid to represent the 100 cookies Corey made. Each square (cookie) can be associated with some ordered pair, where the first component is for the horizontal scale and the second is for the vertical scale.
Use a random digit table (slide 9-60) to create ordered pairs, and then place a tally on the grid to represent each chip.

10. Example (continued)
Estimate the probability that a cookie has exactly one chip by counting the number of squares with exactly one tally and dividing by 100.

11. Computing Odds
Let P(A) be the probability that A occurs and P(A) be the probability that A does not occur. Then the odds in favor of an event A are
and the odds against an event A are

12. Computing Odds In the case of equally like outcomes, odds in favor
odds against

13. Example Find the odds in favor of the event occurring:
a. Rolling a number less than 5 on a die
4 : 2 or 2 : 1
b. Tossing heads on a fair coin
1 : 1

14. Example cont Find the odds in favor of the event occurring:
c. Drawing an ace from an ordinary 52-card deck
4 : 48 or 1 : 12
d. Drawing a heart from an ordinary 52-card deck
13 : 39 or 1 : 3

15. Computing Odds If the odds in favor of event E are m : n, then
If the odds against event E are m : n, then

16. Expected Value
If, in an experiment, the possible outcomes are numbers occurring with probabilities respectively, then the expected value (mathematical expectation) E is given by

17. Expected Value
Expected value can be used to predict the average result of an experiment when it is repeated many times.
Expected value cannot be used to determine the outcome of any single experiment.
Fair game:
When payoffs are involved and the net winnings are $0 (the expected value minus cost to play a game of chance), the game is a fair game.

18. Example
Suppose you pay $5.00 to play the following game. Two coins are tossed. You receive $10 if two heads occur, $5 if exactly one head occurs, and nothing if no heads appear. Is this a fair game? That is, are the net winnings $0?
The net winnings are $0, so this is a fair game.