Question

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

Answer 1

Answer:

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