eBay Scam?

Last night I fell down the YouTube rabbit hole and ended up watching this video. It shows a coin collector who purchased 5 “random” grab bags from an eBay seller. The listing claims the buyer gets a random coin from a lot of 74 total coins. Also, of the 74 coins, 43 of them contain 1 oz of silver (these are worth more). It turned out that the YouTuber only ended up with 1 of the more valuable 1 oz silver coins out of the 5 coins he ordered. That outcome seemed unlikely to him so he had some concerns about whether the eBay seller was running a scam.

Curious about that outcome as well, I decided to calculate the probability of getting exactly 1 of the more valuable coins when ordered 5 total grab bags. For simplicity, I will assume the seller hasn’t sold any grab bags yet so the total number of coins available is still 74. The distribution of the number of 1 oz silver coins selected k at a time without replacement is a hypergeomtric distribution. The probability can be calculated in R like so:

dhyper(1,43,31,5)

## [1] 0.08399124

If we didn’t know this was a hypergeomtric distribution, then we could also estimate this probability using a simple simulation:

set.seed(111)

coins <- c(rep("1oz",43), rep("not_1oz",31))

success <- NULL

for(i in 1:10000){
  bag <- sample(coins, 5, replace = FALSE)
  success[i] <- ifelse(length(which(bag == "1oz")) == 1, 1, 0)
}

mean(success)

## [1] 0.0853

Does an 8% chance indicate a scam?