Question

Consider the experiment consisting of a countably infinite number of Bernoulli trials, where a Bernoulli trial consists of opening a box of macaroni to determine whether a prize is contained within it. Suppose the probability of a box of macaroni containing a prize is p=0.35. Let x be the random variable defined on this sequence of Bernoulli trials that computes the number of macaroni boxes opened until 10 prizes have been

received.
Compute the probability of opening 20 boxes of macaroni to receive 10 prizes.

Answer 1

To compute the probability of opening 20 boxes of macaroni to receive 10 prizes, we use the binomial distribution formula.

To compute the probability of opening 20 boxes of macaroni to receive 10 prizes, we can use the binomial distribution. In this case, a success is defined as finding a prize in a box and a failure is defined as not finding a prize. The probability of a success p is 0.35 and the probability of a failure q is 1 - 0.35 = 0.65. The number of trials n is 20 and the number of successes x is 10.

The probability of exactly x successes in n trials can be calculated using the binomial probability formula:

P(X = x) = nCx * p^x * q^(n-x)

Substituting the given values, we have:

P(X = 10) = 20C10 * 0.35^10 * 0.65^10

Calculating this expression will give us the probability of opening 20 boxes of macaroni to receive 10 prizes.