Consider a binomial probability distribution with p = 0.3 and n = 9. What is the probability of the following?

a) exactly three successes

b) less than three successes

c) seven or more successes

Answer: a) 0.2668, b) 0.4224, c) 0.004

Step-by-step explanation: this is a question under binomial probability.

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

Question a)

n = 9, p = 0.3, q = 1 - p = 1 - 0.3 = 0.7

for our question, r = 3 (exactly 3 successes)

P (x=3) = 9C3 * (0.3) ^3 * (0.7) ^6

P (x=3) = 85 * 0.027 * 0.117649

P (x=3) = 0.2668.

Question b

Probability less that 3 success = p (x<3) = p (x=2) + p (x=1).

At p (x=2)

p (x=2) = 9C2 * (0.3) ^2 * (0.7) ^7

p (x=2) = 36 * 0.09 * 0.0823543

p (x=2) = 0.2668.

At p (x=1)

p (x=1) = 9C1 * (0.3) * (0.7) ^8

p (x=1) = 9 * 0.3 * 0.0576

p (x=1) = 0.1556

p (x<3) = p (x=2) + p (x=1).

p (x<3) = 0.2668 + 0.1556

p (x<3) = 0.4224.

Question c)

P (x≥7) = 1 - p (x≤6)

p (x≤6) can be gotten by using the cumulative binomial table.

From the table, we had that p (x≤6) = 0.996

P (x≥7) = 1 - 0.996

P (x≥7) = 0.004