Possible Statistical Assistance

Question

Could "someone, anyone" tell me what the probabilities are Please!!!!

 

Provide an example of what differentiates a “Probability” from a “Probability Distribution”. In this example identify both the probability and the probability distribution. Suppose we want to determine the (binomial) probability (p) of getting 6 heads in 12 flips of a 2-sided coin. Using the binomial table in the appendix of the text, what values of n, x, and p would we use to look up this probability, and what would be the probability?

Answer 1

The probability is a number. For example, the probability of getting heads in one flip of a fair coin is 1/2

 

A probability distribution, is a function that assigns a probability to every possible event.

Usually, we only define a distribution by specifying the PMF (discrete distributions) or PDF (continuous distributions), since that is sufficient to find the probability of any event. If the distribution is named, like the binomial distribution, we just write down it's name (possibly abbreviated) followed by a list of parameters.

 

As a caution, not all subsets of the sample space need to be considered as being events. In fact, if the support is uncountable, like all real numbers, there are subsets that are non-measurable. Vitali sets are examples of non-measurable sets. See Wikipedia for a simple proof of that. The probability of the r.v. X belonging to such a set is therefore undefined. Thus for continuous distributions, we can't allow all subsets of the sample space to be called events.

 

For example, the Binomial distribution is written Bin(n,p). The interpretation is flipping a coin n times where p is the probability of heads in one flip, and counting the number of heads observed. All flips must be independent.

 

For 12 flips of a fair coin, the number of heads is X ∼ Bin(12, 1/2)

 

So P(X = 6) will be found at n = 12, X = 6, and p = 1/2.