Simulating Experiments Introduction to Random Variable

Simulation The 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 A head or a tail is equally likely to occur 2. 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 coin 2. 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 H/T HHTTHHHTHTTHHHTTTHHHHTTTHHTHHH First round: 2nd round: 3rd round: Yes Yes 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: 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.

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