Susan is taking Western Civilization this semester on a pass/fail basis. The department teaching the course has a history of passing 77% of the students in Western Civilization each term. Let represent the number of times a student takes Western Civilization until the first passing grade is received? (Assume the trials are independent.) a. Write out a formula for the probability distribution of the random variable.

Therefore, how to solve this problem is simple. If you have noticed, the question says, "Let represent the number of times a student takes Western Civilization until the first passing grade is received." The variable that is supposed to go in between the words 'Let' and 'represent' is n.

The problem does not give us the value of n, so we will make one up for the sake of using the formula correctly.

Assume, therefore, that n = 1, it takes the student only one try to pass Western Civilization.
P is automatically assigned a value, which would be .77.

So, using the formula, substituting the values becomes easy.

P(1) = .77(1-.77)^(1-1)
P(1) = .77 or 77%

The random variable in this case is the number of times a student takes Western Civilization until the first passing grade is received. Let's denote this random variable as X.

Since each trial is independent, the probability of passing Western Civilization in any given semester is 77% or 0.77, and the probability of failing is 23% or 0.23.

The probability distribution formula for the random variable X can be written as:

P(X = k) = (1 - 0.77)^(k-1) * 0.77

where k is the number of times a student takes Western Civilization.

This formula calculates the probability that a student takes the course k times and passes on the k-th attempt.

To write out a formula for the probability distribution of the random variable representing the number of times a student takes Western Civilization until the first passing grade is received, we need to consider the probability of passing and failing at each trial.

Let's denote the random variable as X, which represents the number of trials until the first passing grade is received.

Since the probability of passing is given to be 0.77 (77%), and assuming the trials are independent, the probability of failing is the complement of the passing probability, which is 1 - 0.77 = 0.23 (23%).

The probability distribution formula for X can be written as:

P(X = x) = (0.23)^(x-1) * 0.77

This formula calculates the probability of exactly x-1 failures followed by the first passing grade at the x-th trial.

Note that x must be a positive integer (x > 0), as the number of trials cannot be negative or zero.

Therefore, the formula for the probability distribution of the random variable X is P(X = x) = (0.23)^(x-1) * 0.77.

P(n) = p(1-p)^(n-1), where p is the probability of success in each trial, and n is the number of trials.