Probability - Important Concepts and Notes

By Mayank Yadav|Updated : May 16th, 2021

In this article we will provide you you with important notes of Probability. These notes are going to help you in Maths section of state exams as many questions are asked from this section in state exams. Have a look on it. Stay safe, Prepare with BYJU'S Exam Prep.

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BASICS :

We perform an experiment which can have a number of different outcomes. The sample space is the set of all possible outcomes of the experiment. We usually call it S.
For example, if I plant ten bean seeds and count the number that germinate, the sample space is

S = {0,1,2,3,4,5,6,7,8,9,10}.

If I toss a coin three times and record the result, the sample space is

S = {HHH,HHT,HTH,HTT,THH,THT,TTH,TTT},

where (for example) HTH means ‘heads on the first toss, then tails, then heads again.

An event is a subset of S. We can specify an event by listing all the outcomes that make it up. In the above example, let A be the event ‘more heads than tails’ and B the event ‘heads on last throw’. Then

A = {HHH,HHT,HTH,THH}
B = {HHH,HTH,THH,TTH}

Probability = Favorable number of cases/Total Number of cases

Hence if asked what is the probability of event A in above example , then

Probability = n(A)/n(S) = 4/8 = 0.5

We can build new events from old ones:
• A∪B (read ‘A union B’) consists of all the outcomes in A or in B (or both!)
• A∩B (read ‘A intersection B’) consists of all the outcomes in both A and B;
• A\B (read ‘A minus B’) consists of all the outcomes in A but not in B;
• A' (read ‘A complement’) consists of all outcomes not in A (that is, S \A);
• Ø (read ‘empty set’) for the event which doesn’t contain any outcomes

Independent Events :

Two events A and B are said to be independent if

P(A∩B) = P(A)·P(B)

Mutually Exclusive Events :

Two events A and B are said to be mutually exclusive if

P(A∩B) = 0

Collectively Exhaustive Events :

Two events A and B are said to be collectively exhaustive if

P(AUB) = 1

CONDITIONAL PROBABILITY :

Let E be an event with non-zero probability, and let A be any event. The conditional probability of A given E is defined as

P(A | E) = P(A∩E) /P(E)

Example : A random car is chosen among all those passing through Trafalgar Square on a certain day. The probability that the car is yellow is 3/100: the probability that the driver is blonde is 1/5; and the probability that the car is yellow and the driver is blonde is 1/50. Find the conditional probability that the driver is blonde given that the car is yellow.

Solution: If Y is the event ‘the car is yellow’ and B the event ‘the driver is blonde’, then we are given that P(Y) = 0.03, P(B) = 0.2, and P(Y ∩B) = 0.02. So
P(B | Y) = P(B∩Y) / P(Y) = 0.02 /0.03 = 0.667