Lottery Probability

**sahip** · Mar 5th 2010, 04:00 AM

For a particular lottery, the winning numbers are selected by a machine that randomly chooses 5 table-tennis balls from among 45, numbered 1 to 45. The lottery pays off if you match 5, 4, or 3 of the numbers.
a). How many different winning number combinations are there ?
b). What is the probability that you will match all 5 of the winning numbers ?
c). What is the probability that you will match exactly 4 of the 5 winning numbers ?

Thank you.

**Soroban** · Mar 5th 2010, 08:07 AM

Hello, sahip!

These are straight Combinations questions.
Exactly where is your difficulty?

For a particular lottery, the winning numbers are selected by a machine
that randomly chooses 5 table-tennis balls from among 45, numbered 1 to 45.
The lottery pays off if you match 5, 4, or 3 of the numbers.

a) How many different winning number combinations are there?

$_{45}C_5 \:=\:{45\choose5} \:=\:\frac{45!}{5!\,40!} \;=\;1,\!221,\!759$ possible winning combinations.

b) What is the probability that you will match all 5 of the winning numbers?

$P(\text{5 numbers}) \;=\;\frac{1}{1,\!221,\!759}$

c) What is the probability that you will match exactly 4 of the 5 winning numbers?

There are: . $_5C_4 \:=\:{5\choose4} \:=\:5$ ways to match 4 numbers.

Your 5th number is from the other 40 non-winners: . $40$ ways.

. . Hence, there are: . $5\cdot40 \:=\:200$ ways to match exactly 4 numbers.

Therefore: . $P(\text{exactly 4 numbers}) \;=\;\frac{200}{1,\!221,\!759}$

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