Binomial distribution
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This document defines and provides examples of binomial probability distributions. It explains that a binomial probability distribution is a table that lists the possible numbers of successes from n trials paired with each outcome's probability. It also outlines how to calculate the mean, variance, and standard deviation of a binomial distribution. Finally, it demonstrates how to use the distribution table to calculate various probabilities, such as the probability of exactly, at least, more than, or within a range of successes.
Binomial distribution
- 2. Definition A table of all the possible numbers of successes for n number of trials paired with the probability of that number of success.
- 4. StudyTip The sum of the probabilities must equal 1
- 5. Graph The number of successes is the independent variable, and graphed on the horizontal axis The probability is the dependent variable, and graphed on the vertical axis
- 6. Graph The graphs always make this “bell” like shape
- 7. Formula Mean of the distribution: = np Variance of the distribution: 2 = npq Standard deviation of the distribution: = npq
- 8. Example Notice that the mean lies within the most likely outcomes. It is the expected value Notice that the mean lies at the peak of the graph
- 9. Example The standard deviation tells us what is normal. Any event more than 2 standard deviations from the mean is considered highly unlikely. 2 standard deviations from the mean covers the bell part of the graph
- 10. ProblemSolving Use the distribution table to answer questions like: What is the probability of exactly 3 successes? What is the probability of at least 3 successes? What is the probability of more than 3 successes? What is the probability 1 to 4 successes inclusive?
- 11. Example What is the probability of exactly 3 successes? The answer is the probability associated with x = 3 P(x =3) = .298571
- 12. Example What is the probability of at least 3 successes? The answer is the sum of the probabilities for x=3 to x=12 P(x 3) = .298571
- 13. Example What is the probability of more than 3 successes? The answer is the sum of the probabilities for x=4 to x=12 P(x > 3) = .111126
- 14. Example What is the probability 1 to 4 successes inclusive? The answer is the sum of the probabilities from x=1 to x=4 P(1 x 4) = .845512