A new surgical procedure is said to be successful 60% of the time. Suppose the operation is performed six times and the results are assumed to be independent of one another. What are the probabilities of these events? (Round your answers to three decimal places.)

(a) All six operations are successful.

(b) Exactly five are successful.

(c) Less than two are successful.

a) 0.047, b) 0.187, c) 0.041

Step-by-step explanation: the experiment is performed more than once ( six times) and each event are independent on each other, the best distribution that defines the experiment is a binomial probability distribution.

p = probability of success = 60% = 0.6

q = probability of failure = 1 - 0.6 = 0.4

n = number of times experiment was performed = 6

The probability mass function is given as

P(x=r) = nCr × p^r × q^n-r

A)

At x = 6

P(x=6) = 6C6 × (0.6)^6 × (0.4)^0

P(x=6) = 1 × 0.6^6 × 1 = 0.047

B)

At x = 5

P(x=5) = 6C5 × (0.6)^5 × (0.4)^1

P(x=5) = 6 × 0.6^5× 0.4 = 0.187

C)

At x<2

P(x<2) = p(x=1) + p(x=0)

p(x=1) = 6C1 × (0.6)^1× (0.4)^5

P(x=1) = 6 × 0.6^1× 0.4^5 = 0.037

p(x=0) = 6C0× (0.6)^0× (0.4)^6

P(x=6) = 1 × 1 × 0.4^6 = 0.004

P(x<2) = p(x=1) + p(x=0)

P(x<2) = 0.037 + 0.004 =0.041