Question

Consider the experiment of rolling a die 180 times. You win if the number of sixes you roll is at least 27 and at most 34.
Use the normal approximation with continuity correction to estimate the probability of a win.

Answer 1

Using Normal Approximation to Binomial
Mean = n * P = ( 180 * 0.1667 ) = 30.006
Variance = n * P * Q = ( 180 * 0.1667 * 0.8333 ) = 25.004
Standard deviation = √(variance) = √(25.004) = 5.0004

P ( 27 <= X <= 34 )
Using continuity correction
P ( n - 0.5 < X < n + 0.5 ) = P ( 27 - 0.5 < X < 34 + 0.5 ) = P ( 26.5 < X < 34.5 )

X ~ N ( µ = 30.006 , σ = 5.0004 )
P ( 26.5 < X < 34.5 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 26.5 - 30.006 ) / 5.0004
Z = -0.7
Z = ( 34.5 - 30.006 ) / 5.0004
Z = 0.9
P ( -0.7 < Z < 0.9 )
P ( 26.5 < X < 34.5 ) = P ( Z < 0.9 ) - P ( Z < -0.7 )
P ( 26.5 < X < 34.5 ) = 0.8159 - 0.242
P ( 26.5 < X < 34.5 ) = 0.5740