To calculate this, we will use the concept of probability using the binomial distribution.
A random experiment follows a binomial distribution if:
1) The experiment consists of n independent trials.
2) Each trial can result in just two possible outcomes. We call one of these outcomes a success and the other, a failure.
3) The probability of success, denoted by P, is the same on every trial.
Given:
1) The probability of winning a prize (success probability or p) = 0.199.
2) Number of purchases/trials (n) = 13.
3) The number of prizes won (number of successes or k) = 3.
The probability, P(X=k), that an experiment of n trials results in exactly k successes is given by the formula:
P(X = k) = (n choose k) * (p^k) * (1 - p)^(n - k)
Where:
1) (n choose k) represents the number of combinations of n items taken k at a time.
2) p^k is the probability of getting k successes.
3) (1 - p)^(n - k) is the probability of getting n - k failures.
So, we substitute given values into this formula and calculate the probability.
After calculations, the estimated value will be 0.2450, which means there is 24.5% chance that you will win 3 prizes on your 13 eligible purchases.