We roll a die 72 times. Approximate the probability of getting exactly 3 sixes with both the normal and the Poisson approximation and compare the results with the exact probability

The probability of getting exactly 3 sixes using Normal approximation is 0.0022.

The probability of getting exactly 3 sixes using Poisson approximation is 0.0018

Both the probabilities are approximately equal.

Step-by-step explanation:

Let X = outcome of rolling a dice.

The probability of any of the 6 face occurring is, .

The random variable X follows a Binomial distribution with parameters n  and p.

It is provided that the die is rolled n = 72 times.

(1)

As the number of times the die is rolled is quite large the distribution of X can be approximated by the Normal distribution.

Conditions of Normal approximation are satisfied.

So the random variable X follows a Normal distribution with:

Mean = np = 12

Variance = np(1 - p) = 10

Compute the probability of getting exactly 3 sixes as follows:

Thus, the probability of getting exactly 3 sixes using Normal approximation is 0.0022.

(2)

As the ample size is large, i.e. n = 72 > 20 and p = 1/6 = 0.1667 is small a Poisson distribution can be used to approximate the distribution of random variable X.

The probability mass function of Poisson distribution is:

Compute the mean as follows:

μ = np = 12

Compute the probability of getting exactly 3 sixes as follows:

Thus, the probability of getting exactly 3 sixes using Poisson approximation is 0.0018.

The probability of getting exactly 3 sixes in 72 rolls of a die,

When using Normal approximation is 0.0022.

When using Poisson approximation is 0.0018.

Both the probabilities are approximately equal.

addition

step-by-step explanation: