Simulating Experiments

Introduction to Random Variable

SimulationThe imitation of chance behavior

based on a model to accurately reflects

the experiment under consideration

1. State the problem clearly

2. Define the key components

3. State the underlying assumptions

4. Select a model to generate the outcomes for a key components

5. Define and conduct a trial

6. Record the observation of interest

7. Repeats steps 5 and 6 at a large number of times

8. Summarize the information and draw conclusions

Steps in simulating experiments:

Example: A run of three in tossing a coin 10x

State the problem: Toss a coin 10 times. What is the likelihood of a run of at least 3 consecutive heads or 3 consecutive tails

State the assumption: There are 2.1. A head or a tail is equally likely to occur2. Tosses are independent of each other. (what happens on the first toss will not influence the next toss)Assign digits to represent outcomes Using random number table on Table B we assign:1. One digit simulate one toss of the coin2. Odd represents heads, even digits represents tails

Step 1

Step 3

Step 2

Simulate many repetition: looking at 10 consecutive digits on table B, simulate one repetition. Read as many groups of 10 from the table to simulate many repetitions:

Let’s use line 101 of table B for our first three rounds of simulation.

D 1 9 2 2 3 9 5 0 3 4 0 5 7 5 6 2 8 7 1 3 9 6 4 0 9 1 2 5 3 1H/T H H T T H H H T H T T H H H T T T H H H H T T T H H T H H H

First round:2nd round:3rd round:

YesYes

Yes

22 more rounds were added and out of the

25 rounds. 23 of them did have a run

of three

Step 4

State your Conclusion: We estimate the probability of a run of size 3 by the proportion

Estimated probability= 23/25= 0.92

There is a 92% chance of getting a run of three when you toss a coin 10 times.

True mean: 0.826

Step 5

Independent trial: the number of trials has no effect on the succeeding trial

Example: tossing a die, flipping a coin, drawing a card

Dependent trial: shooting 10 free throws in a basketball. Getting an A on the first quiz.

Difference between dependent and independent trial

Shooting free throws

Lisa makes 70% of her free throws in a long season. In a tournament game she shoots 5 free throws late in the game and misses 3 of them. The fans think she was nervous, but the misses may simply be chance. You will shed some light by estimating a probability.

answerShooting Free Throws:

Single random digit will simulate a shot, with 0-6 representing the basket made and 7,8,9 representing the miss.

5 consecutive digits using Table 5 can simulate 5 shots.

After 46 more repetitions:Number of misses 0 1 2 3 4 5

frequency 6 15 18 10 1 0

D 9 6 7 4 6 1 2 1 4 9 3 7 8 2 3 7 1 8 6 8m X X X X X X X x

The relative frequency of missing three or more shots in five attempts is 11/50= 0.22