Thread: Lottery Probability

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.

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?

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

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

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

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

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

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

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

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