Probability: Definition, Formula, Types and Problems

Probability is the branch of mathematics that states how likely an event is to occur in mathematical context. There are events whose outcomes can not be predicted with full certainty. In this case, the best we can say is how likely they are to happen by using the idea of probability.  In this chapter, we will study about the concepts of conditional ptobability of any event given that the other event has occured. It will also explain the probability distribution of random variables and the mean and the variance of probability distribution. 

Read More: Determinant Formula

Key Terms:  Probability, probability of any event, conditional probability, occurrence of event, theorem of Total Probability

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Definition

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The measure of the possibility of any event is known as probability. Event is any phenomenon that happens or is bound to happen in the future. 

The probability line

To understand probability, we need to understand various concepts. Let us know about them in detail.

• Experiment: Any phenomenon that occurs is known as an experiment. Eg - tossing a coin, rolling dice, etc.
• Sample Space: The set of all the possible outcomes of an experiment is known as a sample space.
• Event: The subset of a given sample space is defined as an event. It is associated with a random experiment. An example of an event can be getting head or tail in an experiment of tossing a coin.
• If none of the events is expected to occur in preference to the other, the events are known as equally likely events. For example - when a dice is rolled, all 6 faces of the dice are equally likely to come.
• If the happening of any one event excludes the possibility of occurrence of any other event, the events are known as mutually exclusive events. (A ∩ B) = Φ
• If the performance of an experiment results in the occurrence of at least one of the events, the events are said to be exhaustive events. 
• Any two events are called independent events if the probability of occurrence or non-occurrence of any one event is not affected by the other event
• Outcome: Outcome can be defined as the result of any experiment. Example - getting a six at the roll of a dice.

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Probability of an event

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The probability of happening of any event A can be given by -

P(A) =Number of favourable outcomesTotal number of outcomes= mn

• The probability of any event A is greater than or equal to zero and less than or equal to 1 i.e.

 0 ≤ P(A) ≤ 1.

• The probability of a possible event is 1.
• The probability of an impossible event is 0.

Figure: Probability of events 

• Conditional Probability: If E and F are two events that are associated with a random experiment’s same sample space. Then, the probability of the occurrence of any event E, when the event F has already occurred, is known as the conditional probability of the event E over F. This is denoted by P(E/F).

P(E/F) = P(E

∩F)P(F) (Where P(F)0)

Like this, the conditional probability of event F over E can be given as

P(F/E) = P(E

∩F)P(E)  (where P(E)≠0)

Figure: Conditional probability 

• Properties of Conditional Probability: If E and E are two events of sample space S and G is an event of S which has already occurred such that P(G) ≠ 0, then

(i) P[(E ∪ F)/G] = P(F/G) + P(F/G) – P[(F ∩ F)/G], P(G) ≠ 0

(ii) P[(E ∪ F)/G] = P(F/G) + P(F/G), if E and F are disjoint events.

(iii) P(F’/G) = 1 – P(F/G)

(iv) P(S/E) = P(E/E) = 1

• Multiplication theorem: P (A ∩ B) = P (A). P(B)

For example,in the event of rolling a dice,

Where A is the event of a number less than 4 occurring on a roll of dice, A = {1,2,3}

And B is the event of a multiple of 2 occurring on a roll of dice B ={ 2,4,6}

Hence (A ∩ B) = {2}, and P(A ? B)= 16

• Event(A OR B)

P(A U B) = P(A) + P(B) – P(A ∩ B)

If both A and B are two mutually exclusive events, then P (A ∩ B) = 0 and hence 

P(A U B)= P (A) + P (B)

• Event(NOT A)

If the probability of occurrence of any event A is denoted as P(A), then the probability of the non-occurrence of that same event A is P(A’).

• Event (A BUT NOT B)

If there are two events A and B, then Event (A BUT NOT B) gives us the remainder after the common elements in A and B gets eliminated from A.

• Event (B BUT NOT A)

If there are two events A and B, then Event (B BUT NOT A) gives us the remainder after the common elements in A and B gets eliminated from B.

• Bayes Theorem
• Partition of Sample Space: A set of events E1, E2,…,En is said to represent a partition of the sample space S, if it satisfies the following conditions:

(i) Ei ∩ Ej = Φ; i ≠ j; i, j = 1, 2, …….. n

(ii) E1 ∪ E2 ∪ …… ∪ En = S

(iii) P(Ei) > 0, ∀ i = 1, 2,…, n

• Theorem of Total Probability: Let events E1, E2, …, En form a partition of the sample space S of an experiment.If A is any event associated with sample space S, then

Figure: Division of sample space S

• Bayes Theorem: If E1, E2,…,En are n non-empty events which constitute a partition of sample space S, i.e. E1, E2,…, En are pairwise disjoint E1 ∪ E2 ∪ ……. ∪ En = S and P(Ei) > 0, for all i = 1, 2, ….. n Also, let A be any non-zero event, the probability

To know more about the chapter, 

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Important Topics for JEE Main 

As per JEE Main 2024 Session 1, important topics included in the chapter probability are as follows:

• Formula for Probability
• Types of Probability
• Properties of Probability
• Probability of an Event

Some memory based important questions asked in JEE Main 2024 Session 1 include:

1. Bag A contains 7 white balls & 3 red balls. Bag B contains 3 white balls & 2 red balis. A ball is chosen randomly & found to be red then find the probability that it is taken from bag A.
2. In a paper there are 3 sections A, B and C which has 8, 6 and 6 questions each. A student have to attempt 15 questions such that they have to attempt atleast 4 questions out of each sections, then number of ways of attempting these questions are
3. An urn contains 15 red. 10 white, 60 orange, and 15 green balls. If 2 balls are taken with replacement, find the probability 1 ball is red and the other ball is white.
4. If the mean of 15 observations is 12 and the standard deviation is 3. If 12 is replaced by 10 in data, then the new mean is µ and variance is o² then what is the value of 15(μ + μ² + σ²) = ?