Question

The probability of winning the Powerball jackpot on a single given play is 1/175,223,510. Suppose the powerball jackpot becomes large, and many people play during one particular week. In fact, 180 million tickets are sold that week. Assuming all the tickets are independent of one another, then the number of tickets should be binomially distributed. The values of the parameters n and p in this binomial distribution are:

n=

p=

Then, use the binomial distribution to find the probability that there is one or more winning tickets sold.

______

If X = the number of winning tickets sold, find the mean and standard deviation of the random variable X.

Mean of X =  

Standard deviation of X =  

On the other hand, since the "times" between winning tickets should be independent of one another, the number of winning tickets per week could reasonably be modeled by a Poisson distribution.

The value of the parameter lambda in this Poisson distribution would be _____

The standard deviation of the number of tickets sold in a week (using the Poisson model) is _________

What does the Poisson model predict is the probability of having one or more winning tickets sold? _____

Then, use the binomial distribution to find the probability that there is one or more winning tickets sold. _________

Answer 1

The values of the parameters n and p in this binomial distribution are:

n=180000000

p= 1/175,223,510

Then, use the binomial distribution to find the probability that there is one or more winning tickets sold.

binomial probability is given by

P(X=x) = C(n,x)*px*(1-p)(n-x)

P(X≥1) = 1 - P(X=0) = 1 - C(180000000,0)*(1/175,223,510)^0*(1-1/175,223,510)(180000000) = 1-0.3580 = 0.6420   

If X = the number of winning tickets sold, find the mean and standard deviation of the random variable X.

Mean of X = np =    1.027259413

Standard deviation of X = √(np(1-p))=√1.0273 = 1.0135

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The value of the parameter lambda in this Poisson distribution would be np = 180000000*1/175223510=1.0273


The standard deviation of the number of tickets sold in a week (using the Poisson model) is

std dev = √λ =    1.0135


What does the Poisson model predict is the probability of having one or more winning tickets sold?

poisson probability distribution

P(X=x) = e-λλx/x!

P(X≥1)=1-P(X=0) = 1 - e^-1.01351.0135⁰/0! = 1-0.3580 = 0.6420

Then, use the binomial distribution to find the probability that there is one or more winning tickets sold. _________

binomial probability is given by

P(X=x) = C(n,x)*px*(1-p)(n-x)

P(X≥1) = 1 - P(X=0) = 1 - C(180000000,0)*(1/175,223,510)^0*(1-1/175,223,510)(180000000) = 1-0.3580 = 0.6420