# Probability of winning a lottery

#### Yopi

##### New member
Let's say there are 42 balls and we need to select 5 balls of the winning combinations to win.

The ticket cost €0.4 per 5 ball combination.

And the lottery has a prize structure

If you select all 5 winning balls, you will win € 82,000
for 4 balls, € 110
for 3 balls, € 4.5

So Let's say you buy 5000 tickets
the cost would be € 0.4*5000= € 2000

The probability of all 5 balls selecting the winning one would be 42!/5!*37! = 850668.

So 1 out of 850668.

For 4 balls, 42!/4!*38! = 111930.

So 1 out of 111930.

for 3 balls, 42!/3!*39! = 11480.

So correct me over here... i think i am wrong here

Since we have 5000 tickets.. That means

We have 5000/11480=43.5% chance of getting 3 balls.

so 5000*0.435=2175 tickets of matching 3 balls

Therefore. 2175 x €4.5 = € 9787.5 of winnings?

5000/111930 = 4.45% chance

0.045*5000= 223 tickets matching 4 balls

Therefore, 223 x € 110 = € 24530 of winnings for 4 balls matches?

I am pretty sure i am wrong here and confused. The winnings can't be soo much for just spending € 2000. can someone please explain what am i doing wrong?

#### Jomo

##### Elite Member
For 4 balls, 42!/4!*38! is NOT 111930. You mean for 4 balls, 42!/(4!*38!) = 111930. But I am confused since you have to buy 5 number tickets, not 4 number tickets. Similarly, for 3 balls, it should be 42!/(3!*39!) = 11480. But again, this is the number of ways to buy 3 number tickets.

In fact if you buy a 5 number ticket, then there are 10 different 3 number sets.
For example if you purchase a ticket with the numbers {1, 2, 3 ,4, 5} then the 3 number tickets you have from this is {1,2,3}, {1,2,4}. {1,2,5}, {1,3,4}, {1,3,5}, {1,4,5}, {2,3,4}, {2,3,5}, {2,4,5} and {3,4,5}
You need to go back and start over. If you get stuck then please ask for help.