If 50% of the automobiles sold by an agency for a certain car are equipped with diesel engines, let X represent the number of diesel models among the next 5 cars sold by this agency, write the probability distribution of X.

To write the probability distribution of X, we need to determine the probabilities associated with each possible value of X.

Given that 50% of the automobiles sold have diesel engines, the probability of a car being equipped with a diesel engine is 0.5. Therefore, the probability of having exactly X diesel models among the next 5 cars sold can be calculated using the binomial probability formula, which is:

P(X) = C(n, X) * p^X * (1-p)^(n-X)

Where:
- C(n, X) is the combination of n items taken X at a time (in this case, n = 5 and X varies from 0 to 5),
- p is the probability of success in a single trial (0.5),
- n is the number of trials (5), and
- X represents the number of diesel models (ranging from 0 to 5).

By substituting the values into the formula, we can calculate the probabilities for each value of X:

P(X=0) = C(5, 0) * 0.5^0 * 0.5^(5-0) = 1 * 1 * 0.5^5 = 0.03125

P(X=1) = C(5, 1) * 0.5^1 * 0.5^(5-1) = 5 * 0.5 * 0.5^4 = 0.15625

P(X=2) = C(5, 2) * 0.5^2 * 0.5^(5-2) = 10 * 0.5^2 * 0.5^3 = 0.3125

P(X=3) = C(5, 3) * 0.5^3 * 0.5^(5-3) = 10 * 0.5^3 * 0.5^2 = 0.3125

P(X=4) = C(5, 4) * 0.5^4 * 0.5^(5-4) = 5 * 0.5^4 * 0.5^1 = 0.15625

P(X=5) = C(5, 5) * 0.5^5 * 0.5^(5-5) = 1 * 0.5^5 * 0.5^0 = 0.03125

Therefore, the probability distribution of X for the number of diesel models among the next 5 cars sold by this agency is:

P(X=0) = 0.03125
P(X=1) = 0.15625
P(X=2) = 0.3125
P(X=3) = 0.3125
P(X=4) = 0.15625
P(X=5) = 0.03125