A trainer is teaching a dolphin to do tricks. The probability that the dolphin successfully performs the trick is 35%. If the trainer has the dolphin attempt a trick 20 times, calculate the following probabilities:

a) The probability the dolphin successfully completes the trick 10 out of the 20 attempts.

b) The probability the dolphin successfully completes the trick less than 8 out of the 20 attempts.

c) The probability the dolphin successfully completes the trick at least 6 out of the 20 attempts.

d) The probability the dolphin successfully completes the trick between 6 and 12 times (inclusive) out of the 20 attempts.

To calculate the probability of the dolphin completing the trick for a given number of attempts, we can use the binomial probability formula. b) To find the probability of the dolphin completing the trick less than 8 attempts, we calculate the cumulative probability for X = 0, 1, 2, ..., 7 and sum them up. c) To find the probability the dolphin completes the trick at least 6 times, we calculate the cumulative probability for X = 6, 7, 8, ..., 20 and sum them up.

Explanation:

a) The probability of the dolphin successfully completing the trick 10 out of 20 attempts can be calculated using the binomial probability formula. The formula is P(X = k) = C(n, k) * p^k * (1-p)^(n-k), where n is the number of trials, k is the number of successful trials, p is the probability of success, and C(n, k) is the number of combinations. Substituting the given values, we get:

P(X = 10) = C(20, 10) * 0.35^10 * (1-0.35)^(20-10)
P(X = 10) = 184,756 * 0.35^10 * 0.65^10
P(X = 10) = 184,756 * 0.0008 * 0.0278
P(X = 10) = 4.092

Therefore, the probability the dolphin successfully completes the trick 10 out of 20 attempts is approximately 4.092%.

b) To find the probability the dolphin successfully completes the trick less than 8 out of 20 attempts, we can calculate the cumulative probability for X = 0, 1, 2, ..., 7 and sum them up.

P(X < 8) = P(X = 0) + P(X = 1) + P(X = 2) + ... + P(X = 7)
...

c) To find the probability the dolphin successfully completes the trick at least 6 out of 20 attempts, we can calculate the cumulative probability for X = 6, 7, 8, ..., 20 and sum them up.

P(X >= 6) = P(X = 6) + P(X = 7) + P(X = 8) + ... + P(X = 20)
...

d) To find the probability the dolphin successfully completes the trick between 6 and 12 (inclusive) times out of 20 attempts, we can calculate the cumulative probability for X = 6, 7, 8, ..., 12 and sum them up.

P(6 <= X <= 12) = P(X = 6) + P(X = 7) + P(X = 8) + ... + P(X = 12)
...

Learn more about binomial probability here:

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