Question

5A A Bernoulli Trials experiment consists of 4 trials, with a 4/5 probability of success on each trial.

What is the probability of at least 1 success and at least 1 failure?

What is the probability of 2 successes, given at least 1 success? What is the probability of at least 2 successes, given at least 2 failures?

Enter your answers as whole numbers or fractions in lowest terms.

Answer 1

P(X) = nCx p^x q^n-x

n = 4

p = 4/5

q = 1/5

1. P(at least 1 success and at least 1 failure) = 1 - P(all successes) - P(all failures)

= 1 - (4/5)⁴ - (1/5)⁴

= 368/625

2. P(2 successes | at least 1 success) = P(2 successes) / P(at least one success)

= [4C2 x (4/5)² x (1/5)²] / [1 - (1/5)⁴]

= 2/13

3. P(at least 2 successes | at least 2 failures) = P(2 success and 2 failures) / P(at least 2 failures)

= [4C2 x (4/5)² x (1/5)²] / [P(0 success) + P(1 success) + P(2 success)]

= [4C2 x (4/5)² x (1/5)²]/[(1/5)⁴ + 4 x (4/5)x(1/5)³ + 4C2 x (4/5)² x (1/5)²]

= 96/113