Chapter+5

__**Chapter 5: Binomial Distribution and Poisson Distribution**__
__**Binomial distribution**__ involved surveys with a yes/no type of response and are always conducted under the assumption that you have reliable previously obtained data. So suppose that a given percentage of people feel a certain way about something (based on a previous survey). You now want to know the probability that some number of people out of a given group size will feel the same way. So it's just that, a probability. The formula is somewhat complicated but the tables in the back of the book provide a nice resource.

There is also a formula to predict the outcome (or average) of such a survey. The formula is __**mean = np**__, where n is the number of people being surveyed and p is the base percentage given in the problem. So if 65% of people like green, then out of any given group of 100 people, 65 of them should like green (65%x100 = 65).

You can also calculate the __**standard deviation of the study by finding the square root of npq**__, where q is the percentage of people who disagree with the survey. So continuing from above, we would expect that 65 out of 100 people will like green. The standard deviation is 4.7 (square root of 100x65%x35%). For convenience let's round that to 5. By the empirical rule, that means that in repeated surveys of 100 people at a time, in 68% of the surveys we can expect between 60 and 70 people to like green (± 1 standard deviation). In 95% of the surveys, between 55 and 75 people should like green (± 2 s.d) and in 99.7% of the surveys between 50 and 80 should like green (±3 s.d). The idea would be that it becomes extremely unlikely that you would have a survey result outside of the 50 and 80 marks (less than 50 or more than 80) who like green.

NOTE: If that does happen, it does not mean that we made a mistake. It is however a strong indicator that the original percentage of green lovers was wrong. Less than 50, the percentage was too high; more than 80 it was too low. NOTE: The empirical percentages of 68, 95, and 99.7 are accepted and do not have to be proven.

__**Poisson Distribution**__ works much like binomial but is fundamentally different in terms of the given information. With Poisson, you will be given an average number of occurrences instead of a percentage of occurrences. For example, out of a group of so many people we are told that 9 of them like yellow. So we can predict the probability that out of a different group of people (the same size) only 3 will like yellow. The formula on this one is very complicated so we want the tables for sure.

For Poisson the __**average or expected value is the given value**__ and the __**standard deviation is the square root of that given value**__.

From there it works the same way. Out of a group of 20 people, 9 like yellow. So from a different group of 20 (same size) we would expect 9 to like yellow. The standard deviation is 3 (square root of 9). So in repeated surveys of 20 people, 68% of the time between 6 and 12 should like yellow, 95% should be between 3 and 15 and 99.7% of the time between 0 and 18. You cant have less than 0, but the probability of either 19 or 20 yellow-likers is very low.