Question

The probability that Nick wins a raffle is given by the expression
t
u
.
Write down an expression, in the form of a combined single fraction, for the probability that Nick does not win.

Answer 1

Answer:

The subtlety here is one in language. If you were asked to find the probability that you lose the raffle, you'd be correct, just take 1−P(win). However, you are being asked for the odds against winning.

Odds represent the ratio of possible outcomes, they are different from probability. In simple examples, this roughly means that probability is Number of ways to winTotal number of outcomes, whereas odds against are Number of ways to loseNumber of ways to win.

For example, suppose we have a fair 6-sided die. The odds against you rolling a 2 are 5:1, since there are 5 outcomes in which you lose (fail to roll a two) and only 1 outcome where you win (roll a 2).

In your example, this translates to the following: 9980 of the tickets are other people's tickets; if theirs gets called, you lose. The other 20 are yours. So assuming the lottery is played fairly, the odds against you winning are 9980:20, since in 9980 outcomes, you lose, and in 20 outcomes, you win. This ratio can be reduced to 499:1.