Question

A probability calculator is required on this problem; answer to six decimal places. Suppose we will flip a fair coin 100 times. Using a calculator, find the probability of getting between 42 and 58 heads (inclusive) in two ways: By a Normal approximation: 0.910869 Exactly: 0.920609 How far off is the Normal approximation

Answer 1

• 0.91087
• 0.911374
• 0.000504

Explanation:

Number of toss ( n ) = 100

probability of a head in 1 toss ( p ) = 0.5

Hence : np = 100 * 0.5 = 50 > 5, also n( 1-p) = 50 > 5

mean value ( μ ) = np = 50

std = √ n(p)(1-p) = √100(0.5)(0.5) = 5

Using Normal approximation

P( 42 ≤ x ≤ 58 ) = P ( 42-0.5 ≤ x ≤ 58 + 0.5 )

= P( (41.5 - 50) / 5 < Z < (58.5 - 50) / 5 )

= P ( -1.7 < Z < 1.7 )

= 0.91087

Using Calculator to get the exact value

P( 42 ≤ x ≤ 58 ) = 0.911374

How far off is the Normal distribution (Absolute value of error )= | 0.911374 - 0.91087 | = 0.000504