What are the odds of winning the Mega Millions Jackpot?

Answer 1

The odds of winning prizes are shown in table from the official Mega Millions website. From the table, you can calculate the cumulative probability of winning a prize and the expectation of the prize.

- If you bought 1 ticket

You have 1/75 + 1/141 + 1/844 + 1/306 + 1/13781 + 1/15313 + 1/689065 + 1/3904701 + 1/175711536 = 1/40 chances to win a prize with the expectation of $2/75 + $3/141 + $10/844 + $7/306 + $150/13781 + $150/15313 + $10,000/689065 + $250,000/3904701 + Jackpot/175711536 = $0.83. Here we assumed Jackpot prize is $12,000,000. This can be interpreted statistically as follows: You pay $1 to buy 1 ticket, and you have 2.5% of chances to get $0.83 back out of your $1 and 97.5% of chances to lose your $1. So, the expected net worth of the 1 ticket is 0.025x$0.83=$0.02, which is about 2% of the purchased ticket price.* In case the Jackpot is rolled over, the expectation of the prize will increase, but increasing the Jackpot prize from $12,000,000 to $120,000,000 only results in increasing the expectation of the prize from $0.83 to $1.44, thus the expected net worth of a ticket increases from 0.025x$0.83=$0.02 to 0.025x$1.44=$0.036, which is a trivial difference.

* The calculations are brought from PickSmarter.com.

- If you bought 2 tickets

The chances for at least one of 2 tickets to win a prize is 1-(39/40)2 =0.049 with the expectation of $1.65. So, you pay $2 to buy 2 tickets, and you have 4.9% of chances to get $1.65 back out of your $2 and 95.1% of chances to lose your $2. The expected net worth of the 2 tickets is 0.049x$1.65=$0.08, which is about 4% of the purchased tickets price.

- If you bought 40 tickets

The chances for at least one of 40 tickets to win a prize is 1-(39/40)40 =0.637 with the expectation of $33.04. So, you pay $40 to buy 40 tickets, and you have 63.7% of chances to get $33.04 back out of your $40 and 36.3% of chances to lose your $40. The expected net worth of the 40 tickets is 0.637x$33.04=$21.05, which is about 52% of the purchased tickets price.

- If you bought 100 tickets

The chances for at least one of 100 tickets to win a prize is 1-(39/40)100 =0.92 with the expectation of $83. So, you pay $100 to buy 100 tickets, and you have 92% of chances to get $83 back out of your $100 and 8% of chances to lose your $100. The expected net worth of the 100 tickets is 0.92x$83=$76.4, which is about 76% of the purchased tickets price.

So, the more tickets you buy, the chances of at least one of your tickets win a prize increase with higher expected amount of prize, and thus the chances of your getting nothing decrease, but the amount of money you lose increases when you do lose. The percentage of the expectected net worth of the ticksts gradually converges to the purchased tickets price, at the risk of losing more money with less probability.