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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

3.3 Binomial Distribution

The distribution of the possible number of heads, given N flips with a coin with probability p of flipping heads, is referred to as the Binomial Distribution. It has the form of Equation 3.3, with the “fair coin” probability, 1/2, replaced with p:

P ( h | N , p ) = N ! h ! ( N − h ) ! × p h × ( 1 − p ) N − h (3.3)

Probability of flipping h heads and t tails with an unfair coin Given the probability of flipping a single heads is, say, p and the total number of flips is N=h+t, we have the following equivalent forms:

P ( h , t ) = ( h + t ) ! h ! t ! × p h × ( 1 − p ) t (3.4) P ( h , N ) = N ! h ! ( N − h ) ! × p h × ( 1 − p ) N − h P ( h , N ) = ( N h ) × p h × ( 1 − p ) N − h

where the probability of tails is 1−p.

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 3.2. 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.

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.