What are the odds of winning the Mega Millions jackpot?
It turns out that the odds of winning are 1 in 302,575,350.
Ok, but why? Let's see...
(i). In the Mega Millions game there are 70 white balls, numbered 1 to 70, in one drum. 5 balls are chosen from this drum in the drawing. So there are
70 choose 5 = 12,103,014
ways to choose the 5 white balls. This calculation ignores order, which doesn't matter in Mega Millions. See the image below for a bit more detail.
(ii). We also need to choose the Mega Ball. There are 25 Mega Balls, numbered 1 to 25, in a different drum. 1 ball is chosen from this drum in the drawing. So there are
25 choose 1 = 25
ways to choose the Mega Ball.
(iii). So the number of ways to choose 5 white balls and 1 Mega Ball in the Mega Millions game is
12,103,014*25 = 302,575,350
(iv). Putting this all together we see that the probability of choosing the correct 5 white balls and the Mega Ball, call it p(winning), is
p(winning) = 1/302,575,350 ≈ 3.304961888 × 10^{−9}
Said another way, the odds of winning are 1 in 302,575,350.
#MegaMillions #gamesofchance
References
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[1] "Mega Millions", https://www.megamillions.com/
