Probability Basics

Basic Probability Concepts

Probability measures how likely an event is to happen. It is a number between 0 and 1, inclusive.

• Probability of 0 means the event is impossible.
• Probability of 1 means the event is certain.
• Probabilities between 0 and 1 show how likely or unlikely an event is.

For example, the probability of rolling a 7 on a standard six-sided die is 0 (impossible), while the probability of rolling a number less than 7 is 1 (certain).

The probability scale can be visualised as:

Impossible (0) < Unlikely < Even chance ($0.5$) < Likely < Certain (1)

Events with probability close to 0 are very unlikely to happen, while those close to 1 are very likely.

Calculating Probability

The probability of an event is calculated by dividing the number of favourable outcomes by the total number of possible outcomes, assuming all outcomes are equally likely:

$$\text{Probability} = \frac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}}$$

Probabilities can be expressed as fractions, decimals, or percentages. For example, a probability of $\frac{1}{4}$ is equivalent to 0.25 or $25\%$.

For instance, when tossing a fair coin, there are 2 possible outcomes: heads or tails. The probability of getting heads is:

$$\frac{1}{2} = 0.5 = 50\%$$

Sample Space Diagrams

A sample space is the set of all possible outcomes of an experiment. A sample space diagram lists all these outcomes clearly.

For example, when rolling a six-sided die, the sample space is:

$\{1, 2, 3, 4, 5, 6\}$

Sample space diagrams help visualise all outcomes and are useful for calculating probabilities.

Tree diagrams are another way to represent all possible outcomes, especially for experiments with multiple stages.

For example, tossing two coins can be represented by a tree diagram:

• First toss: Heads (H) or Tails (T)
• Second toss: Heads (H) or Tails (T)

The sample space is: $\{HH, HT, TH, TT\}$, showing all 4 possible outcomes.

Relative and Expected Frequency

When an experiment is repeated many times, the relative frequency of an event is the fraction of times the event occurs. It is an estimate of the probability.

Relative frequency is calculated as:

$$\text{Relative frequency} = \frac{\text{Number of times event occurs}}{\text{Total number of trials}}$$

The more trials there are, the closer the relative frequency tends to get to the true probability (Law of Large Numbers).

Expected frequency predicts how many times an event should occur in a given number of trials, based on the probability:

$$\text{Expected frequency} = \text{Probability} \times \text{Number of trials}$$

For example, if the probability of rain on any day is $0.3$, then in $100$ days the expected number of rainy days is:

$$0.3 \times 100 = 30$$