Trial : Rolling a dice and flipping a coin are trials. A trial is an action which results in one or several outcomes.
Outcome : While flipping a coin we get Head or Tail . Head and Tail are called outcomes. The result of the trial is called an outcome.
Sample point : While flipping a coin, each outcome H or T are the sample points. Each outcome of a random experiment is called a sample point.
Sample space : In a single flip of a coin, the collection of sample points is given by S ={H,T}
If two coins are tossed the collection of sample points S={HH,HT,TH,TT}.
The set of all possible outcomes (or Sample points) of a random experiment is called the Sample space. It is denoted by S.
The number of elements in it are denoted by n(S).
[[Event]] : If a dice is rolled, it shows 4 which is called an outcome (since, it is a result of a single trial). In the same experiment the event of getting an even number is {2,4,6}. So any subset of a sample space is called an event. Hence an event can be one or more than one outcome.
$$ P(A) = \frac{\text{Number of favorable outcomes for event } A}{\text{Total number of possible outcomes}} $$
$$ P(A) = \frac{n(E)}{n(S)} $$ Empirical Approach
- If we are able to predict the exact outcome of an experiment then it is called deterministic experiment.
- If we cannot predict the exact outcome of an experiment then it is called random experiment.
- Sample space S for a random experiment is the set of all possible outcomes of a random experiment.
- An event is a particular outcome or combination of outcomes of an experiment.
- Empirical probability states that probability of an outcome is close to the percentage of occurrence of the outcome.
- If the likelyhood of happening of two events are same then they are known as equally likely events.
- If two events cannot occur simultaneously in single trial then they are said to be mutually exclusive events.
- An event which is sure to happen is called certain or sure event.
![[Screenshot_2024-11-02-01-22-10-531_com.adobe.reader-edit.jpg]]
Addition Theorem of Probability
If A and B are any two events then
- P (A ∪ B) = P(A) + P(B) - P(A ∩ B)
If A,B and C are any three events then
- P(A ∪ B ∪ C) = P(A) + P(B) + P(C) - P(A ∩ B) - P(B ∩ C) - P(A ∩ C) + P(A ∩ B ∩ C )