Probability question! help!?

Question

Betting odds are usually stated against the event happening (against winning).

The odds against event W is the ratio P(notW)/P(W) . In horse racing, the betting odds are based on the probability that the horse does not win. Show that if we are given the odds against an event W as a:b, the probability of not W is
P(not W) = a/a+b

Answer 1

Mathematical probabilities are written as
P(event) = number of times the event occurs/ total number of all possible outcomes

The total number of all outcomes = a plus b (the number of times the horse will win plus the number of times the horse will lose.

The number of times the event occurs (not winning) = a

so P = a/(a+b)

Look at a real example.
If you are given the odds against as 4:1 then there are five possible out comes for five races.
In five races you would expect the horse to not win 4 times and win 1 time.
In probability reasoning we would say
P(horse not winning) = 4/5

Answer 2

The probability that W will happen in x number of tries is given by a/x. and the probability that W will not happen is given as b/x. But since either W will happen or it won't, then x = a+b so probability W is a/(a+b) and probability not W is b/(a+b)

HTH ☺

Doug