Probability

1. Language of Probability

Probability is a measure of how likely an event is to occur. It is expressed as a number between 0 and 1, inclusive, where 0 means the event is impossible and 1 means the event is certain.

• Experiment: A process that can result in one or more outcomes.
• Outcome: A possible result of an experiment.
• Sample Space (S): The set of all possible outcomes of an experiment.
• Event (E): A subset of the sample space.
• Probability of an event (P(E)): The likelihood of an event occurring.

2. Scale of Probability

Probability can be expressed in three ways:

• Fraction: $\frac{\text{Number of favourable outcomes}}{\text{Total number of possible outcomes}}$
• Decimal: The fraction converted to a decimal.
• Percentage: The decimal multiplied by 100.

The probability of any event must be between 0 and 1 (inclusive).

3. Calculation of Probability

The basic formula for calculating probability is:

$$P(E) = \frac{\text{Number of favourable outcomes}}{\text{Total number of possible outcomes}} = \frac{n(E)}{n(S)}$$

Where:

• $P(E)$ is the probability of event E.
• $n(E)$ is the number of outcomes in event E.
• $n(S)$ is the total number of possible outcomes in the sample space S.

Example: A fair six-sided dice is rolled. What is the probability of getting an even number?

Sample Space: $S = \{1, 2, 3, 4, 5, 6\}$

Favourable Outcomes (even numbers): $E = \{2, 4, 6\}$

Number of favourable outcomes: $n(E) = 3$

Total number of possible outcomes: $n(S) = 6$

Probability: $P(E) = \frac{3}{6} = \frac{1}{2} = 0.5 = 50\%$

4. Expected Outcomes

Expected outcome is the average outcome you would expect to get after performing an experiment a large number of times.

For a discrete random variable, the expected value (or mean) is calculated as:

$$E(X) = \sum_{i=1}^{n} x_i P(x_i)$$

Where:

• $E(X)$ is the expected value.
• $x_i$ are the possible outcomes.
• $P(x_i)$ is the probability of each outcome.

Example: A fair coin is tossed twice. The outcomes are represented by H (Head) and T (Tail). What is the expected number of heads?

Possible outcomes: HH, HT, TH, TT

Probability of each outcome: $P(HH) = 0.25$, $P(HT) = 0.25$, $P(TH) = 0.25$, $P(TT) = 0.25$

Expected number of heads: $E(X) = (1 \times 0.25) + (0 \times 0.25) + (0 \times 0.25) + (0 \times 0.25) = 0.25$$

This means that, on average, you would expect to get 0.25 heads for every two coin tosses.

Concept	Definition
Experiment	A process that results in one or more outcomes.
Outcome	A possible result of an experiment.
Sample Space (S)	The set of all possible outcomes.
Event (E)	A subset of the sample space.
Probability of an event (P(E))	The likelihood of an event occurring.