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I’m new to Bayesian statistics (self-studying), R and StackExchange. This forum has been extremely helpful for finding flaws in my reasoning. Is it all right if I ask for help in checking my logic for the question below? Are there any more explanations I should add? My solution matches one of the possible solutions, but I want to know whether my logic is correct and if I am explaining the steps in sufficient detail.

Question:

You decide to conduct a statistical analysis of a lottery to determine how many possible lottery combinations there were. If there are N possible lottery combinations, each person has a 1/N chance of winning. Suppose that 413,271,201 people played the lottery and three people won. You are told that the number of lottery combinations is a multiple of 100 million and less than 1 billion, but have no other prior information to go on. What is the posterior probability that there were fewer than 600 million lottery combinations?

My Solution:

Posterior

= P(Fewer than 600 million lottery combinations | 3 winners out of 413271201 players)

Prior (uniform) = 1/9

There are nine possible outcomes for N. We are counting in increments of 100 million, from 100 million to 900 million. Hence the Prior = 1/9

To find the Likelihood and Data, I use the dbinom function in R to calculate the probability of k successes for a binomial variable, given that the probability of success = 1/N.

Likelihood

= P(3 winners out of 413271201 players | Fewer than 600 million lottery combinations)

= dbinom(3,413271201,1/100000000) +  
dbinom(3,413271201,1/200000000) +  
dbinom(3,413271201,1/300000000) +  
dbinom(3,413271201,1/400000000) + 
dbinom(3,413271201,1/500000000)
= 0.5913931

Data

= 1/9 {
dbinom(3,413271201,1/100000000) + 
dbinom(3,413271201,1/200000000) + 
dbinom(3,413271201,1/300000000) + 
dbinom(3,413271201,1/400000000) + 
dbinom(3,413271201,1/500000000) + 
dbinom(3,413271201,1/600000000) + 
dbinom(3,413271201,1/700000000) + 
dbinom(3,413271201,1/800000000) + 
dbinom(3,413271201,1/900000000)
}
= 0.07351668

Hence, Posterior

= (Prior * Likelihood) / Data
= (1/9 * 0.5913931) / 0.07351668
= 0.894
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    I know this is the correct answer after cross-checking with someone else. This thread can be marked as answered. – 2017-02-18

1 Answers 1

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I argue because the number of players was 413,271,201, We are counting in increments of 100 million, from 500 million to 900 million, not 100 million. Which means that the number of models would be 5. However, I do wonder whether the key distinction is the no other prior information to go on, which would include not knowing how many people participated in the lottery?