There are 49 possible choices for the first ball.
For each of these 49 choices of the first ball, there are 48 possible
choices for the second ball (because one ball has already been taken
out)meaning there are 49 x 48 ways of choosing the first two balls.
However, because the order doesn't matter, this has to be divided by
2 (because the second ball can come before or after the first ball)
Therefore, the chance of a single ticket winning the jackpot is approximately one-in-14 million. Here is another way of looking at it:
If you attempt to guess one number chosen from 49 lottery balls then the probability that you are correct is 1/49. If you choose six numbers then the probability that one of them is the same as the first ball drawn is 6/49.
Because the first ball is not replaced, the probability for choosing the second ball is 1/48, but you have five chances, so the probability for drawing the second number correctly is 5/48.
The probability of choosing all six numbers correctly is:
6/49 x 5/48 x 4/47 x 3/46 x 2/45 x 1/44 = 1/13 983 816.
If your calculator has a button marked nCr you can get the answer directly.
nCr is shorthand for the number of different ways of choosing r items from n where the order does not matter.
"n" stands for the total number (here
n=49)
"r" stands for the number you want to select (here r=6)
So if you
type in the sequence 49 nCr 6 = you should get the result 13 983 816
directly.
Four winning numbers
There are 15 ways to include four of the six winning numbers and 903 ways to include two of the 43 non-winning numbers for a total of 15 x 903 = 13 545 ways to have 4 out of 6 winning numbers, which works out to a probability of 13 545/13 983 816 = 0.0009686, that is odds against of 1 030:1.
Three winning numbers
There are 20 ways to include three of the six winning numbers and 12 341 ways to include three of the 43 non-winning numbers for a total of 20 x 12,341 = 246 820 ways to have 3 out of 6 winning numbers, which works out to a probability of 246 820/13 983 816 = 0,01765, that is odds against of 56:1.
Prize | Odds against | Equivalent number of successive heads |
Jackpot (all 6 numbers correct) | 13 983 816 to 1 | 24 heads in a row |
Third prize (5 numbers correct) | 55 490 to 1 | 16 heads in a row |
Fifth prize (4 numbers correct) | 1 030 to 1 | 10 heads in a row |
Seventh prize (3 numbers correct) | 56 to 1 | 6 heads in a row |