Question

Consider a binomial experiment with 20 trials and probability 0.35 of success on a single trial.

(a) Use the binomial distribution to find the probability of exactly 10 successes. (Round your answer to three decimal places.)

(b) Use the normal distribution to approximate the probability of exactly 10 successes. (Round your answer to three decimal places.)

(c) Compare the results of parts (a) and (b). These results are fairly different. These results are almost exactly the same.

Answer 1

Solution

Given that ,

p = 0.35

1 - p = 0.65

n = 20

x = 10

a)

Using binomial probability formula ,

P(X = x) = ((n! / x! (n - x)!) * p^x * (1 - p)^{n - x}

P(X = 10) = ((20! / 10! (20 - 10)!) * 0.35¹⁰ * (0.65)^{20 - 10}

= 0.0686

Probability = 0.069

b)

According to normal approximation binomial,

X Normal

Mean = = n*P = 20 * 0.35 = 7

Standard deviation = =n*p*(1-p) = 20*0.35*0.65 = 4.55

We using continuity correction factor

P(X = a) = P( a - 0.5 < X < a + 0.5)

P(9.5 < x < 10.5) = P((9.5 - 7)/ 4.55) < (x - ) /  < (10.5 - 7) /4.55 ) )

= P(1.172 < z < 1.641)

= P(z < 1.641) - P(z < 1.172)

= 0.0702

Probability = 0.070

c)

b) These results are almost exactly the same.