Statistics 10 Lecture 11 The Law of Averages
1. Definition of the Law of Averages
Consider a random process in which the probability of success in a
single trial is a fraction, P. A trial might be a single roll of a die, a single flip of a coin, a single spin of a wheel. Suppose that the single trial of this random process is
repeated many times, and that the outcome of each trial is independent of the
others. The larger the number of trials, the more likely it is that the overall
fraction of successes will be close to the probability, P, of success in a single
trial. Also with more trials: You are likely to miss the expected number of outcomes by a larger amount as measured by raw numbers, but you are likely to miss by a smaller amount in terms of percentages.
2. Examples: 10,000 tosses of a coin (handout)
A roulette wheel has 18 black numbers, 18 red numbers and 2 green numbers.
Find the probability of a red on a given spin.
18/38 = 9/19 or about 0.47
If you spin the wheel 3 times, how many times will it come up red?
No clue, it's a random process but you can calculate the chance that it could be 0, 1, 2 or 3 times. HOW?As yourself, how many spins and what are the possible outcomes.
5 Questions you should ask yourself about box models:
1) What numbers (tickets) go into the box?
2) How many of each kind of ticket? You might be thinking in terms of percentages or actually counts or proportions. Example: roulette is a like a box with 3 tickets that are colored red, green, and black and there are 18 red, 2 green, 18 black or about 47.5% red, 5% green, 47.5% black.
3) Should I replace tickets after each draw? In the case of roulette, yes, the spins are independent if you didn't replace the tickets, the odds would change.
4) How many draws? How many times are you going to allow this random process to proceed?
5) What do I do with the numbers I draw? Do I sum them up or calculate an average or calculate a percentage (or proportion)?
I can't stress this enough. Review pages 285-286.