Question

A fair die is tossed 120 times. A success is getting a 2 or a 5 landing face up.

(a) Find the probability of getting exactly 40 successes.

(b) Does the normal approximation to the binomial apply? If so, use it to approximate the probability of exactly 40 successes.

Answer 1

Given that a fair die is tossed 120 times. Now a success is getting a 2 or a 5 landing face up.

Now the probability of getting a 2 or a 5 landing is p=2/6=1/3

Let, X = The number of success in tossing a coin 120 times.

Now what is binomial distribution?

Binomial Distribution

A discrete random variable X is said to have a binomial distribution if its PMF(Probability Mass Function) is given by,

Notation: X~Binomial(n,p)

Mean

Standard Deviation

Coming back to our problem

(a) Here we need to find the probability of getting exactly 40 success.

Hence the probability of getting exactly 40 success is 0.0771

(b) Here we need to see if the normal approximation to the binomial apply and if it applies we need to use it to approximate the probability of exactly 40 successes.

Now,

Hence the normal approximation to the binomial applies.

Now before we go on to use the normal approximation let us know a bit about continuity correction.

Continuity Correction

When a continuous distribution is used to approximate a discrete distribution we use a correction called the continuity correction.

Example:

Normal approximation to the Binomial Distribution

Coming back to our problem

We need to approximate P(X=4) using the normal approximation to binomial.