Respuesta :
Answer:
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, [tex]p=\frac{1}{6}[/tex].
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.
- [tex]np=72\times \frac{1}{6}=12>10[/tex]
- [tex]n(1-p)=72\times \frac{5}{6}=60>10[/tex]
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:
[tex]P(X=3)=P(3-0.5<X<3+0.5)\\=P(2.5<X<3.5)\\=P(\frac{2.5-12}{\sqrt{10}}<\frac{X-\mu}{\sigma}<\frac{3.5-12}{|sqrt{10}})\\=P(-3 < Z<-2.69)\\=P(Z<-2.69)-P(Z<-3)\\=0.00357-0.00135\\=0.00222\\\approx0.0022[/tex]
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:
[tex]P(X=x)=\frac{e^{-\mu}\mu^{x}}{x!};\ x=0,1,2,3...[/tex]
Compute the mean as follows:
μ = np = 12
Compute the probability of getting exactly 3 sixes as follows:
[tex]P(X=3)=\frac{e^{-12}12^{3}}{3!}=\frac{0.01062}{6}=0.0018[/tex]
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.
