- “
**chances for**”

In an experiment where the universe of possible outcomes is made up of equally probable results (random experiment), we determine the “chances for” an event using the ratio:

\(\dfrac{\text{number of favourable results}}{\text{number of unfavourable results}}\)

- “
**chances against**”

In an experiment where the universe of possible outcomes is made up of equally probable results (random experiment), we determine the “chances against” using the ratio:

\(\dfrac{\text{number of unfavourable results}}{\text{number of favourable results}}\)

### Example

Consider the random experiment that consists of rolling an honest die with six faces numbered 1 to 6 and noting the result obtained.

The universe of results is Ω = {1, 2, 3, 4, 5, 6}.

The “chances for” obtaining a 6 are \(\dfrac{1}{5}\) or 1 : 5 while the “chances against” are \(\dfrac{5}{1}\) or 5 : 1.

### Educational Note

The idea of “chance” is familiar to students in all grades. This word is used in many common expressions, such as:

- “I’ve got one chance to do this.”
- “I solved this problem by chance.”
- “If I leave at 1 p.m., I have a better chance of making it to my appointment on time.”
- “When rolling a regular die with 6 faces numbered 1 to 6, I have one chance out of 6 of getting a 4.”
- “I never had a chance.”

This concept belongs to the general idea of opportunity, risk, possibility, or luck. To facilitate an introduction to probability, it’s preferable to use the term “possibility”. That way, we can say that there is one possibility out of 6 for rolling a 6 on an honest die.