Probability

Probability

Probability is a measure of the likelihood that a particular event will occur. When a coin is tossed, we can’t surely say either head or tail appears. At this point, we measure the chance of appearing Head (H) or Tail (T).

Formula for Probability: 

The probability is calculated by taking the ratio of the number of favorable outcomes (Outcome we are interested in) and the total number of possible outcomes. Let us consider number of favorable outcomes is \(n(E)\) and the total number of possible outcomes \(n(S)\). Then, 

Example: If a coin is tossed once, the probabilities of getting a head and a tail are:

\(P(\text{H}) = \frac{1}{2}\) and, \(P(\text{T}) = \frac{1}{2}\)

Note: It ranges from 0 (impossible event) to 1 (certain event).

Mutually Exclusive Events

The events that cannot happen at the same time are called mutually exclusive events. In other words, if one occurs, the other cannot. For example, if we toss a coin, either head (H) or tail (T) appears, not both at the same time. 

Examples: 

1. Tossing a coin – getting heads or tails
2. Rolling a die – getting a 3 or a 5
3. Passing or failing an exam
4. Team winning or losing a match

Addition Law of Probability

The Addition Law of Probability is used to find the probability that at least one of two events occurs.

Statement: 

“If two events \(A\) and \(B\) are mutually exclusive events (cannot happen at the same time), then \(P(A \text{ or } B) = P(A \cup B) = P(A) + P(B)\).

If two events \(A\) and \(C\) are not mutually exclusive events (can happen at the same time), then \(P(A \text{ or } C) = P(A \cup C) = P(A) + P(C) – P(A \cap C)\)”. This is called addition law of probability.

Dependent and Independent Events

1. Dependent Events:

In an experiment, if the occurrence of one event affects the other, then such events are called independent events.

Events are dependent if the outcome of one affects the outcome of the other.

Examples: 

1. Drawing two cards without replacement from a deck.
2. Picking a marble and not putting it back, then picking another.
3. Choosing a student for duty, then choosing another without replacing.
4. Eating one cookie from a jar and picking another one afterward.
5. Removing a bulb from a box and picking the second one.

2. Independent Events:

In an experiment, if the occurrence of one event doesn’t affect the other, then such events are called independent events. 

Note: Events are independent if the outcome of one does not affect the other.

Examples:

1. Tossing a coin and rolling a die.
2. Drawing a card from a deck, replacing it, then drawing again.
3. It rains today and you roll a 5 on a die.
4. A student passes math and another student passes science.
5. Spinning a spinner and flipping a coin.