Basic Probability

Basic Probability

Probability is the measure of how likely an event is to occur, and it ranges from 0 (impossible event) to 1 (certain event). In simple terms, probability quantifies uncertainty or chance.

Basic Probability Formula:

The probability of an event $A$, denoted as $P(A)$, is given by:

$$P(A) = \frac{\text{Number of favorable outcomes for event A}}{\text{Total number of possible outcomes}}$$

Where:

• The favorable outcomes are the outcomes that lead to the event happening.
• The total outcomes are all the possible outcomes in the sample space (the set of all possible outcomes).

The probability value will always satisfy:

$$0 \leq P(A) \leq 1$$

• $P(A) = 0$: Event $A$ cannot happen (impossible event).
• $P(A) = 1$: Event $A$ is certain to happen (certain event).
• $0 < P(A) < 1$: Event $A$ has a chance of happening but is not certain.

Sample Space:

The sample space is the set of all possible outcomes of a random experiment. For example:

• If you roll a six-sided die, the sample space is $\{1, 2, 3, 4, 5, 6\}$.
• If you flip a coin, the sample space is $\{Heads, Tails\}$.

Event:

An event is any subset of the sample space. For example:

• If you roll a die, the event "getting an even number" would be $\{2, 4, 6\}$.

Example 1: Rolling a Fair Die

Suppose you roll a fair six-sided die. What is the probability of rolling a 4?

• Sample space: $\{1, 2, 3, 4, 5, 6\}$
• Favorable outcomes: $\{4\}$
• Total outcomes: 6 (since the die has six sides)

Thus, the probability of rolling a 4 is:

$$P(\text{rolling a 4}) = \frac{1}{6}$$

Example 2: Flipping a Coin

If you flip a fair coin, what is the probability of getting heads?

• Sample space: $\{\text{Heads}, \text{Tails}\}$
• Favorable outcomes: $\{\text{Heads}\}$
• Total outcomes: 2 (since there are two possible outcomes: heads or tails)

Thus, the probability of getting heads is:

$$P(\text{Heads}) = \frac{1}{2}$$

---

Types of Events in Probability

1. Simple Event: A simple event is one that consists of a single outcome. For example, rolling a 3 on a die is a simple event.

2. Compound Event: A compound event is one that consists of two or more simple events. For example, getting a number greater than 4 on a die (which includes the outcomes 5 and 6) is a compound event.

3. Mutually Exclusive Events: Two events are mutually exclusive if they cannot occur at the same time. For example, when you flip a coin, you cannot get both heads and tails at the same time, so the events "heads" and "tails" are mutually exclusive.

4. Non-Mutually Exclusive Events: Two events are non-mutually exclusive if they can occur at the same time. For example, when drawing a card from a deck, the events "drawing a red card" and "drawing a face card" are not mutually exclusive, because there are red face cards (like the Jack of Hearts).

5. Complementary Events: The complement of an event $A$ is the event that $A$ does not happen. The probability of the complement of event $A$ is given by:

   $$P(\text{not A}) = 1 - P(A)$$

   For example, if the probability of raining today is 0.7, then the probability of not raining today is:

   $$P(\text{not raining}) = 1 - 0.7 = 0.3$$

6. Independent and Dependent Events:

   • Independent Events: Two events are independent if the occurrence of one does not affect the probability of the other. For example, flipping a coin and rolling a die are independent events because the outcome of the coin flip does not affect the die roll.
   • Dependent Events: Two events are dependent if the outcome of one event affects the probability of the other. For example, drawing two cards from a deck without replacement is a dependent event because the outcome of the first draw affects the outcome of the second draw.

---

Probability Rules

Addition Rule for Mutually Exclusive Events

For two mutually exclusive events $A$ and $B$, the probability that either event $A$ or event $B$ occurs is the sum of their individual probabilities:

$$P(A \cup B) = P(A) + P(B)$$

Example: If the probability of rolling a 2 is $P(\text{2}) = \frac{1}{6}$ and the probability of rolling a 4 is $P(\text{4}) = \frac{1}{6}$, then the probability of rolling a 2 or a 4 is:

$$P(\text{2 or 4}) = P(\text{2}) + P(\text{4}) = \frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3}$$

Addition Rule for Non-Mutually Exclusive Events

For two non-mutually exclusive events $A$ and $B$, the probability that either event $A$ or event $B$ occurs is given by:

$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$

Where $P(A \cap B)$ is the probability of both events occurring at the same time (i.e., the intersection of events $A$ and $B$).

Example: If the probability of drawing a red card is $P(\text{Red}) = \frac{26}{52}$ and the probability of drawing a face card is $P(\text{Face}) = \frac{12}{52}$, then the probability of drawing a red or a face card is:

$$P(\text{Red or Face}) = P(\text{Red}) + P(\text{Face}) - P(\text{Red and Face})$$

Since there are 6 red face cards, $P(\text{Red and Face}) = \frac{6}{52}$, so:

$$P(\text{Red or Face}) = \frac{26}{52} + \frac{12}{52} - \frac{6}{52} = \frac{32}{52} = \frac{8}{13}$$

Multiplication Rule for Independent Events

For two independent events $A$ and $B$, the probability that both events occur is the product of their individual probabilities:

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

Example: If the probability of flipping heads on a coin is $P(\text{Heads}) = \frac{1}{2}$ and the probability of rolling a 3 on a die is $P(\text{3}) = \frac{1}{6}$, then the probability of flipping heads and rolling a 3 is:

$$P(\text{Heads and 3}) = P(\text{Heads}) \times P(\text{3}) = \frac{1}{2} \times \frac{1}{6} = \frac{1}{12}$$

Multiplication Rule for Dependent Events

For two dependent events $A$ and $B$, the probability that both events occur is given by:

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

Where $P(B | A)$ is the probability of $B$ occurring given that $A$ has already occurred.

Example: In a deck of cards, if you draw one card and do not replace it, the probability of drawing two face cards is:

$$P(\text{Face 1st and Face 2nd}) = P(\text{Face 1st}) \times P(\text{Face 2nd | Face 1st})$$

---

Summary:

• Probability quantifies the likelihood of events, with values between 0 and 1.
• The probability of an event is given by the ratio of favorable outcomes to total outcomes: $P(A) = \frac{\text{favorable outcomes}}{\text{total outcomes}}$.
• Addition Rule: Used to find the probability of the union of two events.
• Multiplication Rule: Used to find the probability of the intersection of two events.
• Complementary Events: The probability of an event not occurring is $P(\text{not A}) = 1 - P(A)$.
• Independent and Dependent Events: The multiplication rule applies differently for these two types of events.