Conditional Probability

Conditional probability is defined as the likelihood of an event based on the occurrence of the previous event. It can be calculated by multiplying the probability of the preceding event by the updated probability of conditional. Ninth-grade students can check the definition, formula, examples and multiplication theorem with proof of the conditional probability on this page.

What Is Conditional Probability?

The probability of an event A is given then another event B occurred is called conditional probability of A given B. It is represented as P(A/B). The formula to find the conditional probability of A given B is here.

P(A/B) = P(A ∩ B)/P(B) and P(B/A) = P(A ∩ B)/P(A)

Example:

Let us say student A is applying for college will be accepted. There is an 80% chance that the individual will be accepted to college. And student B is given dormitory housing. Dormitory housing will be provided for only 60% of all accepted students.

The probability of accepted dormitory housing students is the conditional probability.

P(accepted and dormitory housing) = P(Dormitory housing | accepted) * P(accepted) = (0.60) * (0.80) = 0.48

Conditional Probability Formula

Bayes’ theorem is a formula used in calculating conditional probability. The formulas of conditional probability are mentioned here.

  • The conditional probability of A given B is P(A/B) = P(A ∩ B)/P(B).
  • The conditional probability of B given A is P(B/A) = P(A ∩ B)/P(A).

Proof:

Let us take random experiment E to be repeated N times under identical conditions and A, B are two events connected with E. Suppose, A occurs N(A) times and among these N(A) repetitions another event B also occurs along with A, N(AB) times.

So, N(AB)/N(A) is called the conditional frequency ratio of B on the hypothesis that A has occurred and is denoted by f(B/A). f(B/A) = N(AB)/N(A).

Let, limit n g ∞ f(B/A) exists then this limit is P(B/A). That is conditional probability of B on the hypothesis that A has occurred.

f(B/A) = N(AB)/N(A)

= N(AB)/N/N(A)/N

= f(AB)/f(A)

Therefore, P(B/A) = limit n ∞ f(B/A) = P(AB)/P(A) —- (i)

Provided that P(A) ≠ 0

Similarly if P(B) ≠ P(A/B) = P(AB)/P(B) — (ii)

Provided that P(B) ≠ 0

From (i) and (ii)

P(AB) = P(A/B) . P(B) = P(B/A) . P(A) [Provided P(A) ≠ 0 and P(B) ≠ 0]

Multiplication Theorem of Probability & Proof

In an experiment suppose, A and B are any two events then probabilities of both A and B is given by

P(A ∩ B) = P(A) ∙ P(B/A) ———— (i) or

P(A ∩ B) = P(B) ∙ P(A/B) ———— (ii)

If A and B are independent, then

P(A/B) = P(A) and P(B/A) = P(B)

Substitute P(B/A) = P(B) in (i)

P(A ∩ B) = P(A) . P(B)

Substitute P(A/B) = P(A) in (ii)

P(A ∩ B) = P(B) . P(A) = P(A) . P(B)

If A and B are independent, then probabilities of both A and B is given by

P(A ∩ B) = P(A) . P(B)

Properties of Conditional Probability

Some of the conditional probability properties and their derivations are given here. All these properties depend on the conditional probability formula.

Property I: Let S be the sample space of an experiment and A be the event, then P(S|A) = P(A|A) = 1

Proof:
Using the conditional probability formula
P(S|A) = P(S ∩ A)/P(A) = P(A)/P(A) = 1
P(A|A) = P(A ∩ A)/P(A) = P(A)/P(A) = 1
Hence proved.

Property II: Let S be the sample space of an experiment, A and B be any two events. Let E be any other event such that P(E) ≠ 0. Then P((A U B)|E) = P(A|E) + P(B|E) – P((A ∩ B)|E)

Proof:
Using the formula of conditional probability
P((A U B)|E) = [P((A U B ∩ E)]/P(E)
= [P(A ∩ E) U P(B ∩ E)]/P(E) [using a property sets]
= [P(A ∩ E) + P(B ∩ E) – P(A ∩ B ∩ E)]/P(E) [using addition theorem of probability]
= P(A ∩ E)/P(E) + P(B ∩ E)/P(E) – P(A ∩ B ∩ E)/P(E)
= P(A|E) + P(B|E) – P((A ∩ B)|E) [By conditional probability formula]
Hence proved.

Property III: P(A’|B) = 1 – P(A|B), where A’ si the complement of the set A.

Proof:
Using property 1, P(S|B) = 1
S = A U A’.
So, P(A U A’|B) = 1
Since A and A’ are disjoint events,
P(A | B) + P(A’ | B) = 1
P(A’ | B) = 1 – P(A | B)
Hence proved.

Question’s on Conditional Probability

Question 1:
Two dies are thrown simultaneously, and the sum of the numbers obtained is found to be 7. What is the probability that the number 3 has appeared at least once?

Answer:
Sample space S contains the numbers possible by the combination of two dies.
The combination of outcomes in which 3 has appeared at least once A = {(3, 1), (3, 2), (3, 3)(3, 4)(3, 5)(3, 6)(1, 3)(2, 3)(4, 3)(5, 3)(6, 3)}
The combination of numbers whixh sum up to 7 B = {(1, 6)(2, 5)(3, 4)(4, 3)(5, 2)(6, 1)}
P(A) = \(\frac { 11 }{ 36 } \)
P(B) = \(\frac { 6 }{ 36 } \)
A ∩ B = 2
P(A ∩ B) = \(\frac { 2 }{ 36 } \)
Conditional probability P(A|B) = P(A ∩ B)/P(B) = \(\frac { 2 }{ 36 } \)/\(\frac { 6 }{ 36 } \) = \(\frac { 1 }{ 3 } \)

Question 2:
The probability that it will be sunny on Friday is 4/5. The probability that an ice cream shop will sell ice creams on a sunny Friday is 2/3 and the probability that the ice cream shop sells ice creams on a non-sunny Friday is 1/3. Then find the probability that it will be sunny and the ice cream shop sells the ice creams on Friday.

Answer:
Let us assume that the probabilities for a Friday to be sunny and for the ice cream shop to sell ice creams be S and I respectively. Then,
P(S) = \(\frac { 4 }{ 5 } \)
P(I | S) = \(\frac { 2 }{ 3 } \)
P(I | S’) = \(\frac { 1 }{ 3 } \)
We have to find P(S ∩ I).
We can see that S and I are dependent events. By using the dependent events’ formula of conditional probability,
P(S ∩ I) = P(I | S) · P(S) = (\(\frac { 2 }{ 3 } \)) · (\(\frac { 4 }{ 5 } \)) = \(\frac { 8 }{ 15 } \)
The required probability is \(\frac { 8 }{ 15 } \).

FAQ’s on Conditional Probability

1. What is the conditional probability formula?

The formula of conditional probability is derived from the probability multiplication rule, P(A and B) = P(A) * P(B|A).

2. What is the difference between conditional probability and simple probability?

The main difference between probability and conditional probability is that probability is the likelihood of occurrence of an event A, whereas conditional probability is the probability of an event by assuming a second event has already happened.

3. Define conditional probability with example?

The probability of occurrence of an event A when another event B in relation to A has already occurred is called conditional probability P(A|B). An example is three cards from a deck of 52 cards that are missing. The probability of picking a diamond from the deck can be found using conditional probability.

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