# Probability of winning in the lottery

In the lottery there are 5 numbers rolled from 35 numbers and for 3 right quessed numbers there is a third price. What's the propability that we will win the third price if we buy one ticket with 5 numbers.

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Hi, welcome to math.SE. Please clarify your question as question is not clear.Like what happens when we guess more than $3$ numbers? Are the rolled numbers different? Thanks. – TheJoker Nov 11 '12 at 16:40
Hi, yes they are different and we quess 5 numbers (if at least 3 of them were rolled in lottery we won the third price) – user12392 Nov 11 '12 at 16:44
Check the comment of Ross Millikan, he did the question assuming you need to get exactly $3$ right. Do similarily for exactly $4$ right and exactly $5$ right. Then you get total no of favourable outcomes. Alternatively, find the complementary probability, i.e getting less than $3$ as that would be easier.. – TheJoker Nov 11 '12 at 16:48

## 2 Answers

Hint: There are ${ 35 \choose 5}$ total draws. To get specifically third prize, you need to select $3$ of your $5$ and $2$ of the other $30$.

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OK so I get C(3,5)*C(2,30)/C(5,35) but why do I write *C(2,30)? – user12392 Nov 11 '12 at 16:59
@user12392: I had assumed that if you get four you win second prize and if you get all five you get first prize. But this approach is fine to get those chances, too. I agree with your calculation for three. – Ross Millikan Nov 11 '12 at 17:04
@user12392: for the ${30 \choose 2}$ see André Nicolas' comment to the other answer. He gives some good examples. – Ross Millikan Nov 11 '12 at 18:11

Choose 5 from 35 in $\binom{35}{5}$ and from 5 numbers to get 3 exists $\binom{5}{3}=10$ possibilities and 2 other numbers you choose from 30 others thats not are in your ticket in $\binom{30}{2}=435$ ways so total ways to win third place is $10\times435=4350$ ways.

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Although it is true ${5 \choose 2}={5 \choose 3}$ I think it is easier to see where ${5 \choose 3}$ comes from. – Ross Millikan Nov 11 '12 at 17:02
Why do I choose 2 other numbers from 30 others that are not in my ticket? – user12392 Nov 11 '12 at 17:03
There are many many tickets that will get you a third prize. If the numbers drawn are $1,2,3,4,5$, you will win third prize with $2,3,4, 7,11$ but also with $2,3,4, 12,20$ and with $2,3,4, 31,33$, and so on. – André Nicolas Nov 11 '12 at 17:21
I see. Thank you. – user12392 Nov 11 '12 at 21:58