Probability Basics

 

What Is Probability?

Probability helps us understand how likely something is to happen. We use a number between 0 and 1 to show this likelihood:

• $0$: Impossible
• Between $0$ and $0.5$: Unlikely
• $0.5$: Even chance
• Between $0.5$ and $1$: Likely
• $1$: Certain

You can write probabilities as fractions, decimals or percentages.

 

Key Terms in Probability

• Experiment: An activity with a result, e.g. rolling a dice
• Trial: One go of the experiment
• Outcome: The result of a trial
• Event: One or more outcomes, e.g. "rolling an even number"
• Sample space: The set of all possible outcomes
• Fair event: All outcomes are equally likely
• Biased event: Some outcomes are more likely than others

If we say event $A$ is rolling an even number:

• $A = \{2, 4, 6\}$
• $n(A) = 3$

The probability of event $A$ is written as: $P(A) = \frac{n(A)}{\text{Total outcomes}}$

 

Example: Calculating Basic Probabilities

Question: A bag contains 3 red, 5 blue, and 2 green counters. One counter is picked at random.

(a) Find the probability of picking a blue counter

• Total counters: $3 + 5 + 2 = 10$
• Number of blue counters: 5

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

(b) Find the probability of picking a red or green counter

These events are mutually exclusive (they can't happen at the same time), so:

$P(\text{Red or Green}) = P(\text{Red}) + P(\text{Green}) = \frac{3}{10} + \frac{2}{10} = \frac{5}{10} = \frac{1}{2}$

 

Example

A box contains cards numbered from 1 to 5. Each card is equally likely to be picked.

 

Number	1	2	3	4	5
Probability	?	0.2	0.15	?	0.3

 

(a) Complete the table.

• Add up the given probabilities: $$0.2 + 0.15 + 0.3 = 0.65$$
• The total must be 1, so the missing total is: $$1 - 0.65 = 0.35$$
• We need to share this 0.35 between two outcomes: number 1 and number 4. If they're equally likely: $$\frac{0.35}{2} = 0.175$$

 

Updated table:

Number	1	2	3	4	5
Probability	0.175	0.2	0.15	0.175	0.3

 

(b) Find the probability of picking a number less than 4

That includes numbers 1, 2, and 3: $$P(\text{<4}) = 0.175 + 0.2 + 0.15 = 0.525$$

(c) Find the probability of not picking number 5.

$$P(\text{Not 5}) = 1 - 0.3 = 0.7$$