Maths > Probability > 3.0 Probability
Probability
1.0 Basic Definitions
2.0 Basic Notations
3.0 Probability
4.0 Intersection and Union of Sets of Events
5.0 Conditional Probability
6.0 Multiplication Theorem
7.0 Independent Events
7.1 Occurrence of at least one of the independent events
7.2 Pairwise independent events
7.3 Mutually independent events
8.0 Total Probability Theorem
9.0 Bayes' Theorem
10.0 Illustration for understanding the difference between total probability theorem and baye's theorem
11.0 Probability Distribution of Random Variables
12.0 Probability Distribution
13.0 Mean and variance of a discrete random variable
14.0 Binomial Distribution for Successive Events
15.0 Mean and variance of binomial distribution
3.1 Rule of probability for mutually exclusive events
7.2 Pairwise independent events
7.3 Mutually independent events
The chance of happening of one of them is the sum of the chances of separate events. e.g. if $A$ and $B$ are mutually exclusive, then occurrence of $A$ or $B$ is $$P(A \cup B) = P(A) + P(B)$$.
If ${E_1},{E_2},{E_3},......,{E_n}$ are mutually exclusive events, then $$P({E_1} \cup {E_2} \cup ........ \cup {E_n}) = \sum\limits_{i = 1}^n {P({E_i})} $$
When the events are exhaustive and mutually exclusive, then $$P(A \cup \overline A ) = P(A) + P(\overline A ) = 1$$
Illustration 9. The City Youth Parliament selects one student from a school or none for a given year, for each local division. The probability of selecting a boy from the school is $0.35$, and a girl is $0.45$ and the rest denotes none was selected from the school, for the given year. Find the probability of selecting one student, suppose the committee has decided to select a student from the school. Also find the probability that none is selected.
Solution: Selection of a student i.e. boy or a girl is a mutually exclusive event. And since no one could be selected is all a possibility, selecting a student from the school is not exhaustive.
Thus the required probability is $$0.35 + 0.45$$$$=0.80$$
Now, the event of selecting a student and the event of selecting none is exhaustive an mutually exclusive, total probability is $1$.
Therefore,
$$P(B) = 0.35$$$$P(G) = 0.45$$
and $P(none)$ are exhaustive and mutually exclusive.
i.e. $$ P(B) + P(A) + P(none) = 1$$
Therefore $$P(none) = 1- 0.35 -0.45 = 0.20$$
