Binomial Probability Distributions

Binomial Probability Distribution Requirements

1. The procedure has a fixed number of trial. (A trial is a single observation.) 2. The trials must be independent, meaning that the outcome of any individual trial doesn't affect the probabilities in the other trials. 3. Each trial must have all outcomes classified into exactly two categories, commonly referred to as success and failure. 4. The probability of a success remains the same in all trials.

Using Mean and Standard Deviation for Critical Thinking

A binomial distribution is a particular type of discrete probability distribution, so we could use those same formulas, but if we know the values of n and p, it is much easier to use the following:

Binomial Probability Distribution

Binomial probability distributions allows us to deal with circumstances in which the outcomes belong to two categories, such as heads/tails or acceptable/defective or survived/died.

Using Probabilities to Determine When Results Are Significantly High or Low

Significantly high number of successes: x successes among n trails is significantly high if the probability of x or more successes is 0.05 or less. That is, x is a significantly high number of successes if P(x or more) is less than or equal to 0.05 Significantly low number of successes: x successes among n trails is significantly low if the probability of x or fewer successes is 0.05 or less. That is, x is a significantly low number of successes if P(x or fewer) is less than or equal to 0.05

Range Rule of Thumb

Values are significantly low or high if they differ from the mean by more than 2 standard deviations

5% Guideline for Cumbersome Calculations

When sampling without replacement and the sample size is no more than 5% of the size of the population, treat the selections as being independent (even though they are actually dependent).

Treating Dependent Events as Independent

When selecting a sample (as in a survey), we usually sample without replacement. Sampling without replacement results in dependent events, which violates a requirement of binomial distribution. However, we can often treat the events as if they were independent by applying the 5% guideline.

X and P Must be Consistent

When using a binomial probability distribution, always be sure that x and p are consistent in the sense that they both refer to the same category being called a success.