Question

You roll a fair die 5 times. What is the probability you get at least four 6’s? This time you roll the die 204 times. What is the probability you get between 30 and 40 6’s?

Answer 1

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

P(at least four 6's in 5 rolls) = P(4) + P(5) + P(6)

= 6C4x(1/6)⁴(5/6)² + 6C5x(1/6)⁵x(5/6) + (1/6)⁶

= 0.0080 + 0.0006 + 0.0001

= 0.0087

Norml approximation for binomial distribution

n = 204

p = 1/6

q = 5/6

Mean = np = 34

Standard deviation = = 5.323

P(X < A) = P(Z < (A - mean)/standard deviation)

P(between 30 and 40 6’s) = P(X < 40.5) - P(X < 29.5) (continuity correction of 0.5 applied)

= P(Z < (40.5 - 34)/5.323) - P(Z < (29.5 - 34)/5.323)

= P(Z < 1.22) - P(Z < -0.85)

= 0.8888 - 0.1977

= 0.6911