Question

For a sample size of 6 and a success rate of 40%, find the probability of:

a) 0 successes

b)6 successes

c)at least 5 successes

d)no more than 3 successes

e)fewer than 4 successes

Answer 1

Solution:

n = 6

p = 40% = 0.4

1 - p = 1 - 0.4 = 0.6

X follows the Binomial(n = 6 , p = 0.4)

a)

P(X =0)

= (₆C ₀) * 0.4⁰* (0.6)^{6 - 0}  

= 1 * 0.4⁰ * 0.6⁶

= 0.046656

b)

P(X = 6)

= (₆C ₆) * 0.4⁶* (0.6)^{6 - 6}

= 1 * 0.4⁶ * 0.6⁰

= 0.004096

c)

P(at least 5)

= P(X   5)

= P(X = 5) + P(X = 6)

= (₆C ₅) * 0.4⁵* (0.6)^{6 - 5} + (₆C ₆) * 0.4⁶* (0.6)^{6 - 6}

= 0.036864 + 0.004096

= 0.04096

P(at least 5) = 0.04096

d)

P(no more than 3 successes)

= P(X 3)

= P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)

= 0.046656+0.186624+0.31104+0.27648

= 0.8208

P(no more than 3 successes) = 0.8208

e)

P(fewer than 4 successes)

= P(X < 4)

= P(X 3)

= 0.8208 ..from part d)

P(fewer than 4 successes) = 0.8208