Question

Consider a binomial experiment with 15 trials and probability 0.55 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 almost exactly the same.

These results are fairly different.    

Answer 1

Solution

Given that ,

p = 0.55

1 - p = 0.45

n = 15

x = 10

a)

Using binomial probability formula ,

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

P(X = 10) = ((15! / 10! (15 - 10)!) * 0.55¹⁰ * (0.45)^15-10

=  ((15! / 10! (5)!) * 0.55¹⁰ * (0.45)⁵

= 0.1404

Probability = 0.1404

b)

According to normal approximation binomial,

X $\rightarrow$ Normal

Mean = $\mu$ = n*P = 8.25

Standard deviation = $\sigma$ =$\sqrt{}$n*p*(1-p) = $\sqrt{}$ 15*0.55*0.45 = $\sqrt{}$ 3.7125

We using countinuity correction factor

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

P(9.5 < x < 10.5) = P((9.5 - 8.25)/ $\sqrt{}$ 3.7125) < (x - $\mu$ ) /$\sigma$  < (10.5 - 8.25) / $\sqrt{}$ 3.7125) )

= P(0.65 < z < 1.17)

= P(z < 1.17) - P(z < 0.65)

= 0.8790 - 0.7422

= 0.1368

Probability = 0.1368

c)

The results of parts (a) and (b).

These results are almost exactly the same.