Provide Information Regarding Statistics & Econometrics : Conditional Probability, Dependent & Independent Events Lecture 20

Conditional Probability, Dependent & Independent Events

Lecture 20

Conditional Probability

The probability of an event occurring dependent on the occurrence of a previous event is known as conditional probability.

Let A and B are two events, then the conditional probability of A given B is denoted by:

Similarly, the conditional probability of B given A is calculated as:

Note: If A and B are mutually exclusive events, then A∩B = {} and P(A∩B) = 0.

P(A / B) = 0

P(B / A ) = 0

Example 5.14: Three fair coins are tossed. Find the probability of exactly one head given that at least one head.

Solution: The sample space of three coins toss once is given by:

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

n(S) = 8

Let A represent exactly one head & B represent at least one head.

A = {HTT, THT, TTH}

n(A) = 3

B = {HHH, HHT, HTH, THH, HTT, THT, TTH}

n(B) = 7

A∩B = {HTT, THT, TTH}

n (A∩B ) = 3

The probability of exactly one head given that at least one head is given by:

Example 5.15: If P(A) = 0.4, P(B) = 0.25, and P(A∩B) = 0.2. Find P(A/B) and P(B/A).

Solution: The conditional probability of A given B:

The conditional probability of B given A:

Example 5.16: Given information below:

P(A) = 0.30, P(B) = 0.40 and

Find P(A/B) and P(B/A).

Solution: Using theorem 5.2:

Independent Events

Two or more events are said to be independent events if the probability of an event is not affected by the occurrence of another event. Let A and B are independent events, then mathematically it can be expressed as:

In case of three events

Dependent Events

Two or more events are said to be dependent events if the probability of an event is affected by the occurrence of another event. Let A and B are dependent events, then mathematically it can be expressed as:

In case of three events

The events will be independent if the events are selected without replacement.

Multiplication Law of Probability for Independent events

Statement: If A and B are two independent events, then the probability of A and B is given by:

P(A∩B) = P(A) P(B)

Proof: If A and B events, then the conditional probability of B given A is given by:

P (B/A) = P(A∩B) / P(A) ----------(1)

if A and B are independent, then

P(A∩B) = P(A) P(B)

P (B/A) = P(B)

Equation 1 becomes

P(A∩B) = P(A) P(B)

Multiplication Law of Probability of Dependent Events

Statement: If A and B are two dependent events. Then the probability A and B is given by:

P(A∩B) = P(A)×P(B / A)

Proof: Let S be a sample space having “n” sample points and let A and B be two dependent events. A consists of m1 sample points, and B consists of m2 sample points. Further A∩B consists of m12 sample points.

that's

n (S) = n, n (A) = m1, n (B) = m2, n (A∩B) = m12

Multiplying and divide equation (1) by m1 , we get

Example 5.17: A card is chosen at random from a deck of 52 cards. It is then replaced, and a second card is chosen. What is the probability of choosing a jack and then an eight?

Solution: In this case, both events are independent.

The sample space is given below:

Let A be used to represent jackhead and B be used to represent eight.

The events are independent

Example 5.18: Two cards are selected from a deck of 52 cards. Find the first card is an ace and the second is also an ace.

Considering the following scenarios:.

Scenario 1: If the first selected card had an ace, return the first selected card to the deck and select the second card is also an ace. What is the probability that both are ace?

Scenario 2: The first selected card is set aside, and the second card is also an ace. What is the probability that both are ace?

Solution: In Scenario 1, both cards are selected with replacement, so both events are independent.

Let A be used to represent the first ace and B be used to represent the second ace.

The events are independent

In scenario 2, the first selected card is set aside, and the second card is also an ace. The cards are selected without a replacement method, so the events will be dependent.

Let A be used to represent the first ace and B be used to represent the second ace.