Counting in probability

1. Counting in probability • Permutations • The number of orderings of different events • Combinations • The number ways that outcomes can be grouped

2. Permutations • The number of orderings of different events. • Three cards: AKQ • AKQ, AQK, KQA, KAQ, QAK QKA = 6 • Four digits: 7412 • 24 • Five symbols: # $ % & @ • 120

3. Three different cards: AKQ • AKQ, AQK, KQA, KAQ, QAK QKA = 6 • Complete permutations

4. Partial Permutations • Four Different Digits (of 10) • In the first spot: 10 different digits possible • In the second spot: 9 possible left • In the third spot: 8 possible left • In the last spot: 7 possible left • 10*9*8*7 = 5040

5. Combinations • The number ways that outcomes can be grouped • Calculate the number of combinations of N objects taken r (r <= N) at a time. • Example: 13 cards (AKQ…2) taken 5 at a time = 1287

6. Review the problem • We want to be able to say something about the population from a single sample that we have drawn. • This is the problem of statistical inference • What can we infer from our sample?

7. Our logic for proceeding • If the population has certain characteristics, • then our sample will probably include certain outcomes and probably not include other outcomes • If our sample has outcomes that are unlikely to come from that population, it probably did not come from that population

8. Our logic (continued) • If our sample has outcomes that are unlikely to come from that population, it probably did not come from that population • Our conclusion must be either that the population is as hypothesized or it is not as hypothesized. • We reject or fail to reject the hypothesized population characteristic

9. Once More • We reason from population characteristics to probability distribution of all possible samples -- the sampling distribution. • We calculated this for the population of 7 black and 3 red marbles • We noted that the sampling distribution began to look like a normal distribution

10. Sampling distributions • There are a few sampling distributions that occur very frequently • binomial • normal • student’s t statistic • chi-square statistic • F statistic

11. Binomial • This is the distribution we have already calculated for the red (p=.3) and black (p=.7) marbles. • It is also the distribution for coin tosses where p=.5 for heads and p=.5 for tails • It is the distribution for all binary outcomes.

12. Normal • This is the distribution for many empirical distribution. • This is the distribution which many other distributions approach when working with large numbers: binomial, t, chi-square, etc. • This is the distribution for the sum of a set of random variables.

13. Normal distribution characteristics • The formula

14. Normal distribution characteristics • The shape 68%

15. Normal distribution characteristics • The shape 95%

16. Normal distribution characteristics Mean • The shape One unit is one std. dev.

17. Why the normal distribution? • Central Limit Theorem • Sampling distribution of the mean approaches normal distribution • Mean of sampling distribution approaches mean of population • sd of sampling distribution approaches population sd / sqrt(n) as n becomes large

18. Thus • pop mean = 5, pop sd = 1 • If n=4, sampling dist mean = 5, sd = .5 • If n=9, sampling dist mean = 5, sd = .333 • If n=25, samp dist mean = 5, sd = .20 • If n=900, samp dist mean = 5, sd = .033

19. Pop N=4 N=9 N=900