Question

The chances of winning the Powerball lottery jackpot with a single ticket is 1 in 292,201,228. For the purposes of this problem, assume it is 1 in 300,000,000. During February and March, 210,073,226 Powerball tickets were sold. For the purposes of this problem, assume 200,000,000 were sold.

(a) Assuming the tickets win or lose independently of one another, give the exact probability that there was no jackpot winner during those two months.

(b) Using a basic scientific calculator, give an approximation to your answer in part (a).

You must explain why your approximation is a good one.

PLEASE SHOW ALL WORK.

Answer 1

Answer:

a)

Given,

p = 1 / 300,000,000

n = 200,000,000

consider,

Binomial distribution P(X = r) = nCr*p^r*q^(n-r)

nCr = n!/(n-r)!*r!

P(X = 0) = 200000000C0*(1 / 300,000,000)^0*(1 - 1 / 300,000,000)^(200000000-0)

On solving we get

P(X = 0) = 0.51273301126

b)

Now p = 1 / 300,000,000

q = 1 - p

substitute values

= 1 - 1 / 300,000,000

= 0.99999999666

= 0.999999997 [approximately]

P(No winner) = q^n

substitute values

= 0.999999997^200000000

= 0.5488

Probability = 0.5488

answered by: anonymous