Probability: Equally Likely Outcomes: Quick Revision

A brief note on Probability: Equally Likely Outcomes. Read it before a test. Try each question at the end before you open its answer.

Outcomes and events

• An experiment has results you cannot know in advance. Each result is an outcome.
• An event is a set of outcomes. Its favourable outcomes make it happen.
• Outcomes are equally likely when none has a better chance: a fair die, equal sectors, identical tickets.
• Two or three possible results need not be equally likely, as with win, lose or draw.

The probability of an event

• With equally likely outcomes, $P(E) = \dfrac{\text{favourable outcomes}}{\text{total outcomes}}$.
• $0 \le P(E) \le 1$. An impossible event has $P = 0$; a sure event has $P = 1$.
• The probabilities of all the elementary events add up to $1$.
• For unequal sectors, a sector's chance is its angle out of $360^\circ$.

The complement

• $\text{not } E$ is every outcome that does not favour $E$.
• $P(E) + P(\text{not } E) = 1$, so $P(\text{not } E) = 1 - P(E)$.
• When an event is hard to count, count its complement.

A worked example

One letter is picked at random from SUCCESS. Find $P(\text{S})$ and $P(\text{not S})$.

1. Outcomes. S, U, C, C, E, S, S: $7$ letters.
2. The event. Three S's, so $P(\text{S}) = \dfrac{3}{7}$.
3. The complement. $P(\text{not S}) = 1 - \dfrac{3}{7} = \dfrac{4}{7}$.

Practice with new numbers

A program to try

The program finds three probabilities for one die with exact fractions.

Mistakes to avoid

• Using the formula when the outcomes are not equally likely.
• Dividing by the number of different letters instead of all the letters.
• A probability above $1$ or below $0$: recheck the counts.
• Treating "does not win" as "loses" when a draw is possible.

Check yourself

1. A spinner has nine equal sectors numbered $1$ to $9$. What is the probability of an odd number?

2. What is the probability of a $0$ on one fair die?

3. $P(E) = 0.35$. What is $P(\text{not } E)$?

4. Why is the chance of red on a spinner with a red half and two blue quarters not $\dfrac{1}{3}$?