Mathematics, 23.03.2020 21:36 amakayla57

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

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Answer from: martinezjulio1976

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, $p=\frac{1}{6}$ .

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.

$np=72\times \frac{1}{6}=1210$ $n(1-p)=72\times \frac{5}{6}=6010$

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:

P(X=3)=P(3-0.5

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:

$P(X=x)=\frac{e^{-\mu}\mu^{x}}{x!};\ x=0,1,2,3...$

Compute the mean as follows:

μ = np = 12

Compute the probability of getting exactly 3 sixes as follows:

$P(X=3)=\frac{e^{-12}12^{3}}{3!}=\frac{0.01062}{6}=0.0018$

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.

Answer from: Quest

The answer is -pie over 2

Answer from: Quest

there are 70 more legs than eyes at the dog show (196 – 126 = 70). dogs have two more legs than eyes, and humans have the same number of legs as eyes. thus 70 is twice the number of dogs at the show, so the number of dogs is half of 70, which is 35. to get the number of humans, subtract 35 from 63: 63 – 35 = 28.

step-by-step explanation:

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