Question

Show how to compute the probability of winning the jackpot in the megamillions lottery. The rules are at http://www.megamillions.com/how-to-play under ”How to play”. (a) First let us define an appropriate sample space Ω where Ω = {(i1, i2, i3, i4, i5;i6)|what has to hold about i1, . . . , i6}? (b) How many outcomes are in Ω? (c) What is the probability of winning the jackpot? (d) Do you have a better chance of winning the jackpot in powerball or in megamillions? (e) Consider the probability of flipping a fair coin n times and getting heads every time. (So if n = 3, the probability is 1/8th.) How large does n need to be before the probability becomes smaller than the probability of winning the jackpot in megamillions?

Answer 1

a) In this game, we want to draw six numbers from two separate pools of numbers as (1) five different

numbers from 1 to 70 and (2) one number from 1 to 25 . So we win the jackpot if allthe six selected

numbers match with six winning numbers.

So that i₁, i₂, i₃ , i₄, i₅ are all differents and takes values between 1 to 70.

i₆ is one of the value in between 1 to 25.

Thus the sample space Ω contains all the possible combinations of 1 to 70 and 1 to 25 such that the first five numbers are differents and takes values between 1 to 70. And the last one is from 1 to 25.

So that total combinations = ⁷⁰C₅ * 25 = 12103014 * 25 = 302575350

b) there are total 302575350 outcomes in Ω .

There is only one possible way to select all the matching numbers and a number between 1 to 25.

c) Therefore the probability of winning a Jackpot is 1/302575350 = 0.00000000330496

e) Here we want to find n such that (1/2ⁿ ) < 0.00000000330496 = 3.30496E-09

Let's use iterative procedure in excel:

The formulae used on the above excel sheet are as follws: