Whats the probability with a 3 in 5 chance?

Question

The soda company is printing prizes on the inside of their bottle caps with a 3 in 5 change of winning. If Tatum purchases some soda, what's the probability that the first prize Tatum gets is when she opens the fourth bottle?

The probability that the it takes Tatum less than 5 bottles to win a prize?

The probability that the it takes Tatum at least 5 bottles before she gets a prize?

Answer 1

This is an example of geometric distribution since the chances of winning on any given bottle is constant (3/5) and independent of other bottles and you are calculating the probability of success after a given number of failures.

The probability that Tatum will succeed on the fourth try means that he failed on the first 3 tries. So P(4th try successful) = P(3 failures then 1 success) = (2/5)³ * (3/5) = 0.0384

The probability that Tatum will succeed with less than five attempts means he could win on the first try, the second try, the third try, or the fourth try. If we add up the probabilities of each outcome, we will calculate the probability of less than 5 attempts.

P(less than 5) = P(1st try success) + P(2nd try success) + P(3rd try success) + P(4th try success) = 3/5 + (2/5) * (3/5) + (2/5)² * (3/5) + (2/5)³ * (3/5) = 0.9744

The probability of winning with 5 or more attempts is the complement of winning with less than 5. So P(5 or more attempts) = 1 - P(less than 5) = 1 - 0.9744 = 0.0256