Probability problem: An unbiased die is tossed seven (7) times. Success is recorded only if a 5 or a 6 appears, else it is a failure

Question

a) Determine the probability distribution and list all parameters, together with their respective values.

b) What is the probability of getting exactly three (3) successes?

c) What is the probability that there will be no success at all?

Answer 1

a) Determine the probability distribution and list all parameters, together with their respective values.

b) What is the probability of getting exactly three (3) successes?

probability of one success = probability of rolling 5 or 6 = 2/6 = 1/3

probability of one failure = probability of rolling 1, 2, 3, or 4 = 4/6 = 2/3

Out of 7 tosses, if there are exactly 3 successes, there are also exactly 4 failures. We also need to account for the fact that the successes and failures can occur in any order by using a combination.

probability of exactly 3 successes = ₇C₃ * (1/3)³ * (2/3)⁴ = [7! / (3!*4!)] * (1/3)³ * (2/3)⁴ = (5040/144) * (1/27) * (16/81) = 80640 / 314928 = 0.2561

c) What is the probability that there will be no success at all?

probability of no successes = probability of all failures = (2/3)⁷ = 128 / 2187 = 0.05853