Question

In the Mega Million Lottery, you pick five numbers from 1 – 75 AND one number from 1 – 15.

What is the probability you match ALL the numbers?

Answer 1

Let say Group 1,G1 = 1-75 and Group 2, G2 = 1-15

Total no. of numbers (unordered) possible to pick up = 75C5 X 15C1 ----------------(eqn.1)

Winning combination can be from one of the following cases -

Case 1) 1 no. between 1-15 and 5 between 16-75. No. of favorable cases, say f1=1 (because this one no. can only come from G2)

Case 2) 2 nos. between 1-15 and 4 between 16-75. f2 = 2 (because one of these 2 nos. have come from G2).

Case 3) 3 nos. between 1-15 and 3 between 16-75. f3 = 3 (Similarly any of these 3 can come from G2).

Case 4) 4 nos. between 1-15 and 2 between 16-75. f4 = 4

Case 5) 5 nos. between 1-15 and 1 between 16-75. f5 = 5

Case 6) All 6 nos. between 1-15. f6=6.

Case 7) All 6 nos. between 16-75. f7=0 (since one no. has to be picked from G2 which is between 1 and 15).

Therefore the total no. of favorable cases = summation of f1 to f7 = 1+2+3+4+5+6+0 = 21 ----------------(eqn 2)

Hence the required probability = eqn(2) / eqn (1) = 21 / (75C5 X 15) = 21X8 / (75X74X73X72X71) = 8.1X10^(-8)