A recent market study has determined that the probability that a young adult wil

Question

A recent market study has determined that the probability that a young adult will be willing to try a new online financial service that your company is offering is 50%. In a random sample of 10 young adults, the probability that exactly 5 will be willing to try this new service is _________, and the approximate probability (using the normal distribution is) _________.

0.251; 0.246

0.246; 0.251

0.246; 0.248

0.249; 0.251

0.251; 0.249

0.251; 0.246

0.246; 0.251

0.246; 0.248

0.249; 0.251

0.251; 0.249

Explanation / Answer

Using exact probability,

X ~ Binomial (n,p)

Where n = 10 , p = 0.50

Binomial probability distribution is

P(X) = nCx px (1-p)n-x

So,

P(X = 5) = 10C5 0.505 0.505

= 0.246

By approximate probability using normal distribution,

P( X = x) = P( x - 0.5 < X < x + 0.5) ( Using continuity correction)

P( X < x) = P( Z < x - np / sqrt(np(1-p)) (Using normal approximation)

So,

P( X = 5) = P( 4.5 < X < 5.5)

= P( X < 5.5) - P( X < 4.5)

= P( Z < 5.5 - 10*0.5 / sqrt(10 * 0.5 * 0.5) ) - P( Z < 4.5 - 10*0.5 / sqrt(10 * 0.5 * 0.5) )  

= P( Z < 0.3162) - P( Z < -0.3162)

= P( Z < 0.3162) - (1 - P( Z < 0.3162) )

= 0.6241 - ( 1 - 0.6241)

= 0.248

Exact probability and Approximate probability using normal approximation are 0.246 , 0.248

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