Question

(a) Find the exact probability of exactly 55 heads in 100 tosses of a coin.

(b) Estimate this probability using normal distribution. Compare your answers.

Answer 1

(a)

The exact probability of exactly 55 heads in 100 tosses of a coin = 55/100 = 0.55

(b)

n = 100

p = 0.5

q = 1 - 0.5 = 0.5

= np = 100 X 0.5 = 50

To find P(X=55)

Applying continuity correction, we get:

To find P(54.5 < X < 55.5):

For X = 54.5:
Z = (54.5 - 50)/5

= 0.90

By Technology, Cumulative Area Under the Standard Normal Curve = 0.8159

For X = 55.5:
Z = (55.5 - 50)/5

= 1.10

By Technology, Cumulative Area Under the Standard Normal Curve = 0.8643

So,

P(X=55) = 0.8643 - 0.8159

             = 0.0484

So,

Answer is:

0.0484.

Thus, comparison of answers:

The exact probability of exactly 55 heads in 100 tosses of a coin = 0.55

Estimation this probability using normal distribution = 0.0484

The probability values are completely different because for a continuous distribution the probability value at a particular value of x = 0.