Consider a binomial experiment with n = 6 trials where the probability of success on a single trial is p = 0.05. (Round your answers to three decimal places.)

Question

Consider a binomial experiment with n = 6 trials

where the probability of success on a single trial is p = 0.05.

(Round your answers to three decimal places.)

(a) Find P(r = 0).

(b) Find P(r ≥ 1)

by using the complement rule.

Answer 1

P(r=0) is ₆C₀(0.05)⁰(0.95)⁶ or 6!/0!/(6 – 0)! times 1 times 0.7350918906 equivalent to 0.735.

P(r≥1) is ₆C₁(0.05)¹(0.95)⁵ + ₆C₂(0.05)²(0.95)⁴ + ₆C₃(0.05)³(0.95)³ + ₆C₄(0.05)⁴(0.95)² + ₆C₅(0.05)⁵(0.95)¹ + ₆C₆(0.05)⁶(0.95)⁰.

P(r≥1) is 0.2321342813 + 0.03054398438 + 0.0021434375 + 0.000084609375 + 0.00000178125 + 0.000000015625 equal to 0.2649081094.

P(r≥1) or 0.2649081094 is equal to 1 – P(r=0) or 1 – 0.7350918906 and is equivalent to 0.265.

Answer 2

(a)If the probability of success on a single trial is p = 0.05, then the probability of failing that trial is the complement of p that is 1 - p = 1 - 0.05 = 0.95. If we were to compute the probability that we fail the 6 trails, then

p(r=0) = 0.95 x 0.95 x 0.95 x 0.95 x 0.95 x 0.95 = (0.95)^6 = 0.735091... = 0.735

(b) Using complement's rule: P(r ≥ 1) = 1 - P(r < 1) = 1 - P(r=0) = 1 - 0.735 = 0.265