Probability is defined as the ratio of the number of favorable outcomes to the total number of equally likely possible outcomes of a random experiment.
For example, when we toss a coin, there are two possible outcomes: head or tail. If the coin is fair, the probability of getting a head is 1/2, and the probability of getting a tail is also 1/2.
The following are the fundamental terms needed to understand and solve probability problems.
Random Experiment
A random experiment is an activity or process that produces one outcome from several possible outcomes, but the exact outcome cannot be predicted with certainty before performing it.
Examples:
- Tossing a coin.
- Selecting a ball from a bag containing different colored balls.
Example: When a coin is tossed, the outcome may be Head or Tail. We know the possible outcomes, but we cannot predict the exact result in advance.
Outcome
An outcome is a possible result of a random experiment.
Example:
- When a die is rolled, the possible outcomes are: {1 , 2 , 3 , 4 , 5 ,6}. Each number represents one possible outcome.
- When a coin is tossed, the possible outcomes are Head and Tail.
Sample Space
The sample space is the set of all possible outcomes of a random experiment. It is generally represented by S.
Example: When a coin is tossed twice, the sample space is:
S = {HH, HT, TH, TT}
Event
An event is a set of one or more outcomes of a random experiment. In simple words, an event is the result or condition whose probability we want to find.
Example: When a die is rolled, getting an even number is an event. E = {2,4,6}
Similarly, getting a number greater than 4 is another event: A={5,6}
Favorable Outcome
Favorable outcomes are the outcomes that satisfy the condition of a given event.
Example: When a die is rolled, the favorable outcomes for getting a number greater than 3 are: {4,5,6}
Therefore, the number of favorable outcomes is 3.
Range of Probability
The probability of any event always lies between 0 and 1.
0 ≤ P(E) ≤ 1
Basic Rules
- Addition Rule: P(A∪B) = P(A) + P(B) - P(A∩B), where A∪B denotes the union of events A and B.
- Multiplication Rule for Independent Events: P(A∩B) = P(A) × P(B), where A and B are independent events.
- Complement Rule: P(A′) = 1 - P(A), where ′ A ′ denotes the complement of event A.