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Each month, a state of 4,000,000 citizens is giving an iPhone to 1,000 citizens.

I would like to know the probability of somebody (any single person) winning exactly 2 times in 11 months and the probability of somebody winning exactly 3 times in 11 months.

I've seen the binomial pmf, but I can't figure out where to input the 1,000. Does it go in $p$ (probability of success in one trial) or multiply the result $P$ by 1,000?

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  • $\begingroup$ Hint: what proportion of people win? Protip: Use the normal approximation since the numbers are large. $\endgroup$ Jan 12, 2018 at 19:32
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    $\begingroup$ Please clarify the event of interest: would it be the probability that you win either 2 or 3 times in 11 months (this chance is just a few per million), or is it the probability that somebody wins 2 or 3 times in 11 months (a chance of about 13%)? $\endgroup$
    – whuber
    Jan 12, 2018 at 20:20
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    $\begingroup$ I would like to know the latter $\endgroup$
    – XristosK
    Jan 16, 2018 at 10:40

1 Answer 1

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I did some simple simulations (in R) which can give some indications. But simulating this in R takes quite some time, so to get more replications we need a more efficient implementation ... Following the code:

# For this algorithm,  see B Ripley Stochastic Simulation page 80
reservoir_sampling <- function(N, pods) {
    stopifnot(N > pods)
    csample <- 1:pods
    for (seen in seq(from=pods+1, to=N, by=1))if(runif(1)<=(pods/seen))csample[sample(1:pods, 1)] <- seen
    return(csample)
}

sim_one <- function(N=4000000, k=11, pods=1000) {
    winners <- matrix(0L, nrow=k, ncol=pods)
    for (round in seq(from=1, to=k, by=1)) winners[round, ]<-reservoir_sampling(N,  pods)
    wintab <- table(c(winners))
    names(wintab) <- NULL
    wintab <- factor(wintab, levels=1:k)
    wintab <- table(wintab)
    wintab
    }

SIMS <- replicate(1000, sim_one() )   

In this 1000 replications I get somebody winning twice in ALL the replicas, somebody winning thrice in 11 replicas, so quite far from @whubers value in comment of about 13%.

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