Chapter 2: Conditional Probability

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Chapter 2: Conditional Probability

Conditional Probability

Let A and B be two events of finite sample space U. The probability that B occurred given that A has occurred is denoted by P(B/A) and is called conditional probability.

Theorem 1 If A and B are any two events of a finite sample space U, then show that

          

Theorem 2 Multiplication Theorem of Probability or Theorem of Compound Probability:

If the probability of an event A happening as a result of trial is P(A) and after A has happened that probability of an event B happening as a P(B/A), then the probability of both the events A and B happening as a result of two trial is P(AB) or

OR

Independent Events: If neither of two events A and B affects the probability of happening of the other, we say that they are independent.   

Remarks: 1. The event A and B are independent if             

             

Otherwise A and B are dependent.

2. If A and B are independent, then

                                          

Example 1 Suppose a packet of 10 razor blades has 2 defective blades in it. Two blades are drawn from the packet one after another without replacement. Find the probability that both blades drawn are defective.

Solution: Let E₁ be the event that first blade drawn is defective and E₂ the event that second blade drawn is defective. Here E₁ and E₂ are independent events and                        

Hence                 

Example 2 Two cards are drawn from a bridge deck, without replacement. What is the probability that the first is an ace and the second is a king?

Solution:

Example 3: An urn contains 10 black and 10 white balls. Find the probability of drawing two balls of the same colour.

Ans: 9/19

Example 4: A bag contains four white and two black balls and a second bag contains three of each colour. A bag is selected at random, and a ball is then drawn at random from the bag chosen. What is the probability that the ball drawn is white?

Ans 7/12

Example 5:  Three machines I,II and III manufacture respectively 0.4,0.5 and 0.1, of the total production. The percentage of defective items produced by I, II and III is 2, 4 and 1 percent respectively. For an item chosen at random, what is the probability it is defective ?

Ans : 0.029

Example 6 Five salesmen A, B, C, D and E of a company are considered for a three member trade delegation to represent the company in an international trade conference. Construct the sample space and find the probability that:

(i)                 A is selected.

(ii)             A is not selected, and

(iii)         Either A or B (not both) is selected.

(Assume the natural assignment of probability)

 Solution: The sample space for selecting three salesmen out of 5 salesmen A, B, C, D and E for the trade delegation is given by:

S= {ABC, ABD, ABE,}