Pick 10 Lottery For the New York Pick 10 lottery, the player f...

Question

Pick 10 Lottery For the New York Pick 10 lottery, the player first selects 10 numbers from 1 to 80. Then there is an official drawing of 20 numbers from 1 to 80. The prize of $500,000 is won if the 10 numbers selected by the player are all included in the 20 numbers that are drawn. Find the probability of winning that prize.

Accepted Answer

All right, hello, everyone. So this question says, in a lucky draw, a player selects 7 distinct numbers from 1 to 50. Then 10 numbers are randomly drawn from the same set of 1 to 50 during the official drawing. To win the jackpot, all 7 of the players' numbers must be among the 10 numbers drawn. What is the probability of winning the jackpot? Option A says about 0. B says 0.01, C says 0.031, and D says 0.12. OK, so first and foremost, we're looking for the probability of winning this jackpot, given that each player draws 7 distinct numbers, and all 7 have to be amongst the 10 drawn in order to actually win. So, the total number of outcomes, just in general, would be. All the possible combinations of 10 numbers. Ranging from 1 to 50 But We can also distinguish between the number of favorable outcomes, not just the total number of outcomes. And for this question, we're more interested in the favorable outcomes because we're looking for the chances of actually winning. So to find the total number of favorable outcomes, we want to find the total number of 10 number sets, ranging from 1 through 50 that contain all 7 of the players' numbers. Now, if the player has already picked 7 numbers, the other 3 have to be drawn. To create the full 10 number drop. So, the 3 numbers that are remaining have to come from the remaining 43 numbers, because that's 43 or that's 50 rather, subtracted by the 7 that the player already picked. And then As mentioned before, 3 numbers are remaining to create the full set. So that's 43 C3. So now the probability of a win. Is equal to the number of favorable outcomes divided by the number of total outcomes. So that's 4310 or 3, 43 C3. Divided by 50 C 10. And evaluating this expression gives you 1.2. To the power of 10 to -6. Or rather 1.2 multiplied by 10-6 power. And this Is equal to approximately 0. Which means our correct answer choice is option A. And the odds are definitely stacked against the player of this game. But there you have it. So with that being said, thank you so very much for watching, and I hope you found this helpful.

Key Concepts

Combinatorics

Probability

Hypergeometric Distribution

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