Probability & Statistics 1 (S1)

Probability: Rules, Conditional Probability, Mutually Exclusive and Independent Events

1. Basic Probability Rules

Probability is a measure of the likelihood of an event occurring. It is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.

• Definition: The probability of an event A occurring, denoted by $P(A)$, is calculated as: $$P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$$
• Complement Rule: The probability that an event does not occur is the complement of the event. $$P(A^c) = 1 - P(A)$$
• Addition Rule (for mutually exclusive events): If events A and B are mutually exclusive (they cannot occur at the same time), then: $$P(A \cup B) = P(A) + P(B)$$
• Addition Rule (for non-mutually exclusive events): If events A and B are not mutually exclusive, then: $$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$
• Multiplication Rule (for independent events): If events A and B are independent, then: $$P(A \cap B) = P(A) \times P(B)$$
• Multiplication Rule (for dependent events): If events A and B are dependent, then: $$P(A \cap B) = P(A) \times P(B|A)$$ where $P(B|A)$ is the conditional probability of B given A.

2. Conditional Probability

Conditional probability is the probability of an event occurring given that another event has already occurred. It is denoted by $P(A|B)$, which reads "the probability of A given B".

The formula for conditional probability is:

$$P(A|B) = \frac{P(A \cap B)}{P(B)}$$

Example: Consider a bag with 5 red balls and 3 blue balls. If a ball is drawn at random and it is blue, what is the probability that it is red?

Here, A is the event that a red ball is drawn and B is the event that a blue ball is drawn. We want to find $P(A|B)$.

The probability of drawing a blue ball is $P(B) = \frac{3}{8}$. The probability of drawing a blue ball and then a red ball is $P(A \cap B) = 0$ (since a blue ball cannot be red). Therefore, $P(A|B) = \frac{0}{3/8} = 0$.

3. Mutually Exclusive Events

Events A and B are mutually exclusive if they cannot both occur at the same time. In other words, $P(A \cap B) = 0$.

Example: Consider rolling a fair six-sided die. Let A be the event that an even number is rolled, and B be the event that a number greater than 4 is rolled. These events are mutually exclusive because the only even number greater than 4 is 6, and a die cannot show both an even number and a number greater than 4 simultaneously.

4. Independent Events

Events A and B are independent if the occurrence of one event does not affect the probability of the other event occurring. The formula for the probability of two independent events is:

$$P(A \cap B) = P(A) \times P(B)$$

Example: Consider flipping a fair coin twice. Let A be the event that the first flip is heads, and B be the event that the second flip is tails. These events are independent because the outcome of the first flip does not influence the outcome of the second flip.

Event Type	Definition	Formula
Complement	The event does not occur.	$P(A^c) = 1 - P(A)$
Mutually Exclusive	Events cannot occur at the same time.	$P(A \cup B) = P(A) + P(B)$
Independent	The occurrence of one event does not affect the probability of the other.	$P(A \cap B) = P(A) \times P(B)$