The probability that Roberto throws a baseball in the strike zone on one pitch is 0.72. Roberto throws a baseball 3 times in a row.
Create a probability distribution for X, the number of times Roberto throws the baseball in the strike zone. Show how you obtained each probability. Round each probability to three decimal places.
Find the probability that Roberto throws at least one of the baseballs in the strike zone. Show your work or explain your answer.
To create a probability distribution for X, we need to calculate the probability of each possible outcome, from X = 0 to X = 3 (since he throws the ball 3 times). We do this using the binomial probability formula: P(X = k) = C(n, k) * p^k * (1-p)^(n-k), where C(n, k) is the combination of n items taken k at a time, p is the probability of success (in this case, throwing the ball in the strike zone), and n is the number of trials.
1. P(X = 0).
In this case, Roberto does not throw the ball in the strike zone at all (3 failures). Using the binomial probability formula with n = 3, k = 0, and p = 0.72, we have:
P(X = 0) = C(3, 0) * 0.72^0 * (1-0.72)^(3-0) = 1 * 1 * 0.28^3 ≈ 0.022
2. P(X = 1).
In this case, Roberto throws the ball in the strike zone once (1 success, 2 failures). Using the binomial probability formula with n = 3, k = 1, and p = 0.72, we have:
P(X = 1) = C(3, 1) * 0.72^1 * (1-0.72)^(3-1) = 3 * 0.72 * 0.28^2 ≈ 0.216
3. P(X = 2).
In this case, Roberto throws the ball in the strike zone twice (2 successes, 1 failure). Using the binomial probability formula with n = 3, k = 2, and p = 0.72, we have:
P(X = 2) = C(3, 2) * 0.72^2 * (1-0.72)^(3-2) = 3 * 0.72^2 * 0.28 ≈ 0.504
4. P(X = 3).
In this case, Roberto throws the ball in the strike zone all three times (3 successes). Using the binomial probability formula with n = 3, k = 3, and p = 0.72, we have:
P(X = 3) = C(3, 3) * 0.72^3 * (1-0.72)^(3-3) = 1 * 0.72^3 * 1 ≈ 0.373
Now we have the probability distribution for X: X | P(X)
0 | 0.022
1 | 0.216
2 | 0.504
3 | 0.373
To find the probability that Roberto throws at least one of the baseballs in the strike zone, we can find the complement: P(X ≥ 1) = 1 - P(X < 1) = 1 - P(X = 0) ≈ 1 - 0.022 = 0.978. So the probability is approximately 0.978.