What is the exact probability that John will pass three out of his next four SMAD tests?

To find the exact probability that John will pass three out of his next four SMAD tests, we can apply the binomial probability formula, which is suitable for scenarios involving a fixed number of independent trials, each with two possible outcomes (in this case, passing or failing a test).

The binomial probability formula is:

[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} ]

Where:

• ( P(X = k) ): the probability of k successes in n trials

• ( n ): total number of trials (in this case, 4 tests)

• ( k ): number of successful outcomes we want (in this case, passing 3 tests)

• ( p ): probability of success on an individual trial

• ( \binom{n}{k} ): the binomial coefficient, calculated as ( \frac{n!}{k!(n-k)!} )

To solve the problem, we need the probability of passing a single test, denoted as ( p ). Assuming that John has a 70% chance (or 0.7) of passing a single test, the probability of failing (1-p)