| 5.2 | Estimating Probabilities | >
- Key Idea 1: The five-step simulation approach that we used for estimating expected values can be used for estimating probabilities...
- Consider the problem "In a family of three children, what is the probability that all three are boys?" Assume that boys and girls have an equal chance of being born.
- Here is a procedure.
Discussion
- Key Idea 2: How many trials should one carry out?
- For 100 trials, 2/3 of the time the estimated probability will be within 0.05 of the true theoretical probability.
- For 400 trials, 2/3 of the time the estimated probability will be within 0.025 of the true theoretical probability.
- One always has the option to do more simulations (especially with computers).
| Discussion |
|
|
- Key Idea 3: Alternative ways of defining the test statistic in "Step 3" of the six-step procedure...
- We could assign a "1" to a successful trial and a "0" to an unsuccessful trial; in this case Step 5 merely becomes the computation of the average of the statistic of interest.
- Illustration with
Box Sampler. - We could simply count the number of "tails;" in this case Step 5 becomes a count of how many trials yielded a 3.
- Illustration with Box Sampler.
- Key Idea 4: A slight modification in the question being asked requires a corresponding modification in the simulation procedure...
- Suppose we ask "In a family of three children, what is the probability that at least two are boys?"
- We can use the same simulation that we just did (the last one in Key Idea 3), changing step 5 so that it is a count of how many trials yielded a 2 or more.
- Illustration with Box Sampler.
- Key Idea 5: Of course, not all problems can be solved using a coin as a model for Step 1...
- Suppose the first traffic light on your route to class is green for 20 seconds and red for 40 seconds.Thus, the light has 60-second cycle. What is the probability that you will get exactly three green lights on the next four mornings?
- A coin with 50/50 probability of landing heads or tails will not do the job.
- Here is a procedure.
| Discussion |
|
|
| Discussion |
|
|
| Discussion |
|
|
- Key Idea 6: In chapters 8 and 14 we will learn how to get theoretical solutions to these binomial problems...