Question

The coach says that the probility of winning the next game is 40%. what are the odds that team will win?

Answer 1

The odds are `2:3`.

Explanation:

There is a small but important difference between probability and odds.

A probability is a ratio of "the outcome we're interested in" to "all possible outcomes". It measures, out of all possible things that could happen, what fraction of them are ones we would call a "success".

`"Probability = " "number of outcomes we want"/"all outcomes"`

An odds is a ratio of "the outcome we're interested in" to "everything else". It compares the number of ways a thing could happen to the number of ways that thing could not happen.

`"Odds = " "successful outcomes"/"not successful outcomes"`

Ever hear someone say the phrase "fifty-fifty"? That's an odds. They're saying that, out of 100 trials, 50 would be counted as "success", and the other 50 would not.

We can find an odds from a probability very easily. Imagine there are `N` things that could happen. Then if `p` of those things are "successes", the probability of success is `p/N,` while the odds of success are `p/(N-p).` (If `p` things are "good", then the remaining `N-p` things are "not good".)

Quick solution:

`"Prob." = p/N = 40% = 4/10`

`=>"Odds" = p/(N-p) = 4/(10-4)= 4/6 = 2/3 = 2:3`

Long solution:

In this question, we know `p` is 40% of `N.` So `N=p/(40%)=2.5p.`

Substituting this into the odds formula gives:

`"Odds " = p/(N-p) = p/(2.5p - p) = cancelp/(1.5cancelp) = 1/1.5 = 2/3 = "2 : 3"`

Recall: an odds is mathematically different from a probability. To help convey this visually, we often write an odds using colon notation instead of as a fraction.