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

The probabilities are listed below:

If each trial represents an independent event, then the probability of success of a given number of consecutive trials () in defined by the following formula:

(1)

Where:

If we know that , then the probabilities associated to a given number of trials:

1) 3 successes

The probability of success of exactly 3 successes is 5.27 %.

2) 6 successes

The probability of success of exactly 6 successes is 0.28 %.

3) 1 success

The probablity of success of exactly 1 success is 37.5 %.

4) 5 successes

The probablity of success of exactly 5 successes is 0.74 %.

5) At least 1 success

The probability of success of at least 1 success is at most 37.5 %.

6) At least 2 successes

The probability of success of at least 2 successes is at most 14.1 %.

7) At least 6 successes

The probablity of success of at least 6 successes is at most 0.28 %.

8) At least 7 successes

The probability of success of at least 7 successes is at most 0.10 %.

9) At most 7 successes

The probability of success of at most 7 successes is at least 0.10 %.

10) At most 6 successes

The probability of success of at most 6 successes is at least 0.28 %.

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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

answer: expected value of x = 1.0

step-by-step explanation:

p(x=0) = 0.25

p(x=1) = 0.50

p(x=2) = 0.25

expected value for x = 0×0.25 + 1×0.5 + 2×0.25 = 1.0

take the logarithm of both sides (the base of the log doesn't matter):

apply the exponent property - this says that :

distribute the log terms on both sides:

group like terms together:

divide through both sides by the coefficient on :

you can do some additional rewriting here using the properties of the logarithm to simplify things if you like: