# Lottery Probability

• May 7th 2010, 07:49 PM
sahip
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.
What is the probability that you will match exactly 4 of the 5 winning numbers ?

Thank you.
• May 8th 2010, 12:33 AM
CaptainBlack
Quote:

Originally Posted by sahip
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.
What is the probability that you will match exactly 4 of the 5 winning numbers ?

Thank you.

Suppose you match balls 1 to 4 but not 5, the probability of this is:

$\frac{1}{45}\times \frac{1}{44} \times \frac{1}{43}\times \frac{1}{42}\times \frac{40}{41}$

(this is the probability of matching ball 1, 2, 3, 4 and not matching ball 5)

Now the final probability is $5$ times this (the unmatched ball can be any of the five)

CB
• May 8th 2010, 03:40 AM
undefined
Quote:

Originally Posted by CaptainBlack
Suppose you match balls 1 to 4 but not 5, the probability of this is:

$\frac{1}{45}\times \frac{1}{44} \times \frac{1}{43}\times \frac{1}{42}\times \frac{40}{41}$

(this is the probability of matching ball 1, 2, 3, 4 and not matching ball 5)

Now the final probability is $5$ times this (the unmatched ball can be any of the five)

CB

Shouldn't you additionally multiply by 5! to account for the fact that the balls could be picked in 5! different orders?

If so, then the answer would agree with an alternate approach I tried:

Without loss of generality, we can suppose that the balls numbered 1 through 5 were randomly chosen, and we want the probability that exactly four match.

{1,2,3,4,x}
{1,2,3,x,5}
{1,2,x,4,5}
{1,x,3,4,5}
{x,2,3,4,5}

The above is to illustrate that there are 40*5 combinations for which exactly four match, so the probability is given by 40*5/C(45,5).
• May 8th 2010, 03:52 AM
CaptainBlack
Quote:

Originally Posted by undefined
Shouldn't you additionally multiply by 5! to account for the fact that the balls could be picked in 5! different orders?

If so, then the answer would agree with an alternate approach I tried:

Without loss of generality, we can suppose that the balls numbered 1 through 5 were randomly chosen, and we want the probability that exactly four match.

{1,2,3,4,x}
{1,2,3,x,5}
{1,2,x,4,5}
{1,x,3,4,5}
{x,2,3,4,5}

The above is to illustrate that there are 40*5 combinations for which exactly four match, so the probability is given by 40*5/C(45,5).

Yes.

CB
• May 8th 2010, 04:00 AM
Quote:

Originally Posted by sahip
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.
What is the probability that you will match exactly 4 of the 5 winning numbers ?

Thank you.

Alternatively,

there are $\binom{5}{4}$ ways to choose 4 of the 5 winning numbers.

The selection of 5 must contain one of the 40 remaining numbers,

hence there are $\binom{5}{4}\binom{40}{1}$ ways to match 4.

There are $\binom{45}{5}$ ways to choose 5 from 45

so the probability of matching 4 of the 5 winning numbers with a single selection is

$\frac{\binom{5}{4}\binom{40}{1}}{\binom{45}{5}}$