The number of failures is k = X - 4.
k follows a negative binomial distribution: Negative binomial distribution - Wikipedia, the free encyclopedia
p = 18/38 = 9/19 and the number of successes is r = 4.
.
Therefore
A casino patron will continue to make $5 bets on red in roulette until she has won 4 of these bets.
On each bet, she will either win $5 with probability or lose $5 with probability .
If W is her final winnings and X is the number of bets she makes, then, since she would have won 4 bets and lost (X-4) bets, it follows that
Hence
but why is ?
The number of failures is k = X - 4.
k follows a negative binomial distribution: Negative binomial distribution - Wikipedia, the free encyclopedia
p = 18/38 = 9/19 and the number of successes is r = 4.
.
Therefore