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📚 Statistical Inference for Everyone
Chapters ▾
1 Introduction to Probability2 Applications of Probability3 Random Sequences and Visualization4 Introduction to Model Comparison5 Applications of Model Comparison6 Introduction to Parameter Estimation7 Priors, Likelihoods, and Posteriors8 Common Statistical Significance Tests9 Applications of Parameter Estimation and Inference10 Multi-parameter Models11 Introduction to MCMC12 Concluding Thoughts13 Appendix A: Computational Analysis14 Appendix B: Notation and Standards15 Appendix C: Common Distributions and Their Properties16 Appendix D: Tables

15.4 Binomial

Binomial distribution The discrete binomial distribution is defined to be the probability of achieving h successes in a given N events where each event has a given θ probability of success.

P ( h | N , θ ) = ( h N ) θ h ( 1 − θ ) N − h

Three dotted probability curves for the number of heads in 30 flips under different coins: p=0.1 peaks near 3 heads (about 0.24), p=0.5 peaks at 15 heads (about 0.145), and p=0.8 peaks near 24 heads (about 0.18). The three bells barely overlap.
Figure C.4. Probability of getting h heads in 30 flips given a possible unfair coin. One coin has p=0.1, where the maximum is for 3 heads (or 1/10 of the 30 flips), but 2 heads is nearly as likely. Another has p=0.5, and is the fair coin considered earlier with a maximum at 15 heads (or 1/2 of the 30 flips). Finally, another coin shown as p=0.8 where 24 heads (or 8/10 of the 30 flips) is maximum.

Although it may look like a Beta, the binomial distribution is used to find the best estimate for the number of successes, h, given the number of events, N, and the probability of the success of a single event, θ.

Adapted from Statistical Inference for Everyone, by Brian Blais (Bryant University), licensed under CC BY-SA 4.0 (dual-licensed under the GNU FDL 1.2 or later; this adaptation uses the CC BY-SA grant). Changes were made; this adaptation is distributed under the same license. License: CC-BY-SA-4.0.