Consider a state lottery where you get to choose 8 numbers
from 1 to 80, no repetition allowed. The Lottery Commission
chooses 11 from those 80 numbers, again no repetition. You
win the lottery if at least 7 of your numbers are there in
the 11 chosen by the Lottery Commission.
What is the probability of winning the lottery?
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The probability of winning the lottery is two in one billion
i.e. only two person can win from one billion!!!
Let's find out sample space first. The Lottery Commission
chooses 11 numbers from the 80. Hence, the 11 numbers from
the 80 can be selected in 80C11 ways which is very very high
and is equal to 1.04776 * 1013
Now, you have to select 8 numbers from 80 which can be
selected in 80C8 ways. But we are interested in only those
numbers which are in 11 numbers selected by the Lottery
Commision. There are 2 cases.
? You might select 8 numbers which all are there in 11
numbers choosen by the Lottery Commission. So there are 11C8
ways.
? Another case is you might select 7 lucky numbers and 1
non-lucky number from the remaining 69 numbers. There are (
11C7 ) * ( 69C1 ) ways to do that.
So total lucky ways are
= ( 11C8 ) + ( 11C7 ) * ( 69C1 )
= (165) + (330) * (69)
= 165 + 22770
= 22935
Hence, the probability of the winning lottery is
= (Total lucky ways) / (Total Sample space)
= (22935) / ( 1.04776 * 1013)
= 2.1889 * 10-9
i.e. 2 in a billion.
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