Probability is a way of measuring the chances of something happening. It is measured by the ratio of the favourable events to the whole number of possible events.
When provided with a sample space (S) and an event (E), the probability of an event E occurring is denoted by P(E).
NOTE:
- The sum of the probabilities of all elementary events in an experiment is 1. (The sum of probability of occurrence of an event and the probability of occurrence of the complement of that event is always 1.)
- P(E) + P( E′) = 1
- Probability is always between 0 and 1.
- 0 ≤ P(E) ≤ 1
- If it can't happen at all, then the probability is 0. (When the occurrence of an event is 0, the event is impossible.)
- P(E) = 0
- If we are sure something will happen, the probability is 1. (When the occurrence of an event is 1, the event is certain.)
- P(E) = 1
Terms Related to Probability
Experiment : An experiment is any activity associated with a certain result. For example:
- Tossing a coin.
- Throwing a die.
- Selecting a card.
Random Experiment : A random experiment is like a test or trial where we can't be sure what will happen. A trial is an action that will result in one or several outcomes. For example:
- Flipping a coin will result in two outcomes.
- Rolling a fair six-sided die will result in six outcomes.
Outcomes :An outcome refers to a possible result or consequence of a random experiment. For example:
- Flipping a coin will result in two outcomes - Heads and tails.
- Rolling a six-sided dice is a random experiment. The possible outcomes for this experiment are the numbers 1 through 6.
If we consider the event of rolling a number 3, then the outcome “3” is one of the possible outcomes of this experiment.
Sample Space: The sample space is all the possible outcomes we could get from a random experiment. It is denoted by S. For example:
- If a coin is tossed, then the sample space would be either getting heads (H) or tails (T).
- If we roll a die, then the sample space would be the numbers 1, 2, 3, 4, 5 or 6.
Event: An event for an experiment is the collection of some outcomes of the random experiment. It is denoted by E. For example:
- Getting a head on tossing a coin.
- Getting a face card when a card is drawn from a pack of 52 cards.
Complementary Event: A complementary event for an experiment represents the non-occurrence of the outcomes with respect to sample space. If event E is the occurrence of a particular outcome, then the complementary event is denoted as E'. For example: Consider the tossing of a fair six-sided dice. Total outcomes like 1, 2, 3, 5 and 6.
- If E is the event of rolling a dice to get 4, then the complementary event E′ would be the event of not rolling a dice to get 4.
- If E is the event of rolling a dice to get 5, then the complementary event E′ would be the event of not rolling a dice to get 5.