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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

2.6 The Lottery Problem or Rare Things Are Common

This problem is identical to the birthday problem mathematically,with the only difference that the probability numbers are much smaller and the number of participants is much larger. We start with a story about someone winning the lottery twice in the same day!

Pretty amazing! That's something like

P ( winning two tickets ) = 1 2 · 10 13 ∼ 5 · 10 − 14 (2.3)

which truly is quite improbable as a single event, but is it truly an improbable event to happen somewhere? The assumption stated in the quote is that only two tickets were purchased. We all know that many lottery tickets are purchased daily, which should increase the chance that somewhere this will occur. Even this winning couple purchased tickets every day for 20 years before winning this.

Like the birthday problem, you have to set up the problem in the negative, and as what the probability of no one winning two lotteries. If we assume 5 million people playing daily for 20 years, this probability is:

P ( no one winning two tickets | 5 million plays daily for 20 years ) = ( 1 − 1 2 · 10 13 ) 5 · 10 6 × 365 × 20 (2.4) ∼ ( 1 − 5 · 10 − 14 ) 5 · 10 6 × 365 × 20 = 0.998 (2.5)

yielding a 0.2% chance of this happening sometime in those 20 years - still pretty rare, but not outrageously so. If we further imagine that this is occurring across the 50 states, this increases to 10% chance of this happening sometime in those 20 years. If we further imagine that there are as many as 5 different lotteries (there are usually more) that could be played per state, this jumps up to 40%.

What we see as an initially highly unlikely event starts to become likely and in fact common when considering all of the possible ways that event could be produced.

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.