An experiment consists of 8 independent trials where the probability of success on each trial is 3 8. Find the probability of obtaining the following. Round answers to the nearest ten-thousandth. 13. Exactly 3 successes. 14. Exactly 6 successes. 15. Exactly 1 success. 16. Exactly 5 successes. 17. At least 1 success. 18. At least 2 successes. 19. At least 6 successes. 20. At least 7 successes. 21. At most 7 successes. 22. At most 6 successes

Step-by-step explanation:

This is a binomial distribution because the probabilities are either that of success or failure.

If probability of success, p = 3/8 = 0.375, then probability of failure, q = 1 - p = 1 - 0.375 = 0.625

The formula is expressed as

P (x = r) = nCr * p^r * q^ (n - r)

Where

x represent the number of successes.

p represents the probability of success.

q = represents the probability of failure.

n represents the number of trials or sample.

n = 8

13) P (x = 3) = 8C3 * 0.375^3 * 0.625^ (8 - 3) = 0.28

14) P (x = 6) = 8C6 * 0.375^6 * 0.625^ (8 - 6) = 0.03

15) P (x = 1) = 8C1 * 0.375^1 * 0.625^ (8 - 1) = 0.11

16) P (x = 5) = 8C5 * 0.375^5 * 0.625^ (8 - 5) = 0.1

17) P (x ≥ 1) = 1 - P (x < 1)

P (x < 1) = P (x = 0)

P (x = 0) = 8C0 * 0.375^0 * 0.625^ (8 - 0) = 0.023

P (x ≥ 1) = 1 - 0.023 = 0.977

18) P (x ≥2) = 1 - P (x < 2)

P (x < 2) = P (x = 0) + P (x = 1)

P (x = 0) = 8C0 * 0.375^0 * 0.625^ (8 - 0) = 0.023

P (x = 1) = 8C1 * 0.375^1 * 0.625^ (8 - 1) = 0.11

P (x ≥ 2) = 1 - (0.023 + 0.11) = 0.867

19) P (x ≥ 6) = P (x = 6) + P (x = 7) + P (x = 8)

P (x = 6) = 8C6 * 0.375^6 * 0.625^ (8 - 6) = 0.03

P (x = 7) = 8C7 * 0.375^7 * 0.625^ (8 - 7) = 0.005

P (x = 8) = 8C8 * 0.375^8 * 0.625^ (8 - 8) = 0.0004

P (x ≥ 6) = 0.03 + 0.005 + 0.0004 = 0.0354

20) P (x ≥ 7) = P (x = 7) + P (x = 8)

P (x ≥ 7) = 0.005 + 0.0004 = 0.0054

21) P (x ≤ 6) = 1 - P (x = 8)

P (x ≤ 6) = 1 - 0.0004 = 0.9996

22) P (x ≤ 6) = 1 - [P (x = 7) + P (x = 8) ]

P (x ≤ 6) = 1 - (0.005 + 0.0004) = 0.9946