Basic Probability

Spec reference: Probability
Key idea: Calculate probabilities, use probability notation and understand experimental vs theoretical probability.

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▶ Sets and Venn Diagrams

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Probability scale

Probability is measured from 0 (impossible) to 1 (certain).

$$P(\text{event})=\frac{\text{number of favourable outcomes}}{\text{total number of outcomes}}$$

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Basic probability

Example

A bag contains 3 red, 5 blue and 2 green balls. A ball is picked at random.

$P(\text{red})={\displaystyle \frac{3}{10}}$

$P(\text{blue})={\displaystyle \frac{5}{10}}={\displaystyle \frac{1}{2}}$

$P(\text{green})={\displaystyle \frac{2}{10}}={\displaystyle \frac{1}{5}}$

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Complementary events

$$P(\text{event does NOT happen})=1-P(\text{event happens})$$

Example

$P(\text{rain})=0.35$

$P(\text{no rain})=1-0.35=\boxed{{\displaystyle 0.65}}$

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Mutually exclusive events

Events that cannot happen at the same time.

$$P(A\text{ or }B)=P(A)+P(B)$$

For all mutually exclusive outcomes: $\sum P=1$

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Expected frequency

$$\text{Expected frequency}=P(\text{event})\times \text{number of trials}$$

Example

A biased coin has $P(\text{heads})=0.6$. In 200 flips, how many heads are expected?

$0.6\times 200=\boxed{{\displaystyle 120}}$

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Relative frequency (experimental probability)

$$\text{Relative frequency}=\frac{\text{number of times event occurred}}{\text{total number of trials}}$$

The more trials, the closer relative frequency gets to the true theoretical probability.

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Sample space diagrams

List all possible outcomes to find probabilities.

Example

Two dice are rolled. Find $P(\text{sum}=7)$.

The sample space has $6\times 6=36$ outcomes. Pairs that sum to 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) = 6 outcomes.

$P(\text{sum}=7)={\displaystyle \frac{6}{36}}=\boxed{{\displaystyle {\displaystyle \frac{1}{6}}}}$

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Exam tips

Watch out for

• Probabilities must be between 0 and 1 inclusive
• All probabilities for mutually exclusive exhaustive events must add to 1
• Theoretical probability assumes equally likely outcomes; experimental probability is based on results

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Test Yourself

Question 1 of 5

Bag: 4 red, 6 blue, 2 green. P(red)?