A trainer is teaching a dolphin to do tricks. The probability that the dolphin successfully performs the trick is 8 out of 24 attempts. Let X = the number of times the dolphin successfully performs the trick.
Write the formula for the binomial distribution
What is the probability, to three decimals, that the dolphin succeeds 15 times
What is the probability, to 3 decimals, the the dolphin will succeed at most 6 times
The formula for the binomial distribution is:
P(X=k) = (n choose k) * p^k * (1-p)^(n-k)
where P(X=k) is the probability of getting k successes, n is the number of trials or attempts, p is the probability of success in each trial, (n choose k) is the number of ways to choose k successes from n trials (calculated as n! / (k! * (n-k)!)), and ^ denotes exponentiation.
To find the probability that the dolphin succeeds 15 times, we can use the formula:
P(X=15) = (24 choose 15) * (8/24)^15 * (1-8/24)^(24-15)
Calculating this probability to three decimal places, we get:
P(X=15) ≈ 0.228
Therefore, the probability that the dolphin succeeds 15 times is approximately 0.228.
To find the probability that the dolphin succeeds at most 6 times, we need to calculate the probability of getting 0, 1, 2, 3, 4, 5, or 6 successes and then sum them up. Using the binomial distribution formula:
P(X≤6) = P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4) + P(X=5) + P(X=6)
To calculate this probability to three decimal places, we add up the probabilities:
P(X≤6) = (24 choose 0) * (8/24)^0 * (1-8/24)^(24-0) + (24 choose 1) * (8/24)^1 * (1-8/24)^(24-1) + (24 choose 2) * (8/24)^2 * (1-8/24)^(24-2) + (24 choose 3) * (8/24)^3 * (1-8/24)^(24-3) + (24 choose 4) * (8/24)^4 * (1-8/24)^(24-4) + (24 choose 5) * (8/24)^5 * (1-8/24)^(24-5) + (24 choose 6) * (8/24)^6 * (1-8/24)^(24-6)
Calculating this probability to three decimal places, we get:
P(X≤6) ≈ 0.035
Therefore, the probability that the dolphin will succeed at most 6 times is approximately 0.035.