We have studied what probability is and how it can be measured. We dealt with simple problems. Now we shall consider some of the laws of probability to tackle complex situation. There are two important theorems, viz., (1) the Addition Theorem and (2) the Multiplication Theorem.
Addition Theorem : The simplest and most important rule used in the calculation is the addition rules, it states, “If two events are mutually exclusive, then the probability of the occurrence of either A or B is the sum of the probabilities of A and B. Thus, P(A or B)=P(A)+P(B)
Example 9 :
A bag contains 4 white, 3 black and 5 red balls. What is the probability of getting a white or a red ball at random in a single draw ?
Solution : The probability of getting a white ball = 4/12
The probability of getting a red ball =
The probability of a white or a red
When events are not mutually exclusive
The addition theorem studied above is not applicable when the events are not mutually exclusive. In such cases where the events are not mutually exclusive, the probability is : P(A or B) = P(A) + P(B) - P(A and B)
Example 10 :
Two students X and Y work independently on a problem. The probability that A will solve it is 3/4 and the probability that Y will solve it is 2/3. What is the probability that the problem will be solved ?
P(A or B)=P(A)+P(B)-P(A and B)
The probability that X will solve the problem is = 3/4 The probability that Y will solve the problem is—2/3 The events are not mutually exclusive as both of them may solve the problem.
Therefore, the probability =
Alternatively: The probability that X will solve it and Y fail to solve it = 3/4 x 1/3 = 3/12
∴ Probability that the problem will be solved
Alternatively : The probability that X will fail to solve and will Y solve it
= 1/4 x 2/3 = 2/12
∴ Probability that the problem will be solved =
Alternatively : The probability that neither X nor Y will solve it
Hence, the probability that the problem will be solved
Multiplication: When it is desired to estimate the chances of the happening of successive events, the separate probabilities of these successive events are multiplied. If two events A and B are independent, then the probability that both will occur is equal to the product of the respective probabilities. We find the probability of the happening of two or more events in succession.
P(A and B)=P(A)xP(B)
Example 11 : In two tosses of a fair coin, what are the chances of head in both ?
Solution : Probability of head in first toss = 1/2
Probability of head in the second toss = 1/2
Probability of head in both tosses = 1/2x1/2 -1/4
Example 12 :
The probability that X and Y will be alive ten years hence is 0.5 and 0.8 respectively. What is the probability that both of them will be alive ten years hence ?
Solution : Probability of X being alive ten years hence = 0.5
Probability of Y being alive ten years hence = 0.8
Probability of X and Y both being alive ten years hence =.5x.8=0.4
When events are dependent : If the events are dependent, the probability is conditional. Two events A and B are dependent ; B occurs only when A is known to have occurred.
P (B|A) means the probability of B given that A has occurred.
Example 13 :
A man want to marry a girl having qualities: White complexion the probability of getting such girl is 1 in 20.
Handsome dowry - the probabihty of getting is 1 in 50. Westernised style - the probability is 1 in 100.
Find out the probability of his getting married to such a girl, who has all the three qualities.
The probability of a girl with white complexion or 0.05. The probability of a girl with handsome dowry = or 0.02. The probabihty of a girl with westernised style = or 0.01. Since the events are independent, the probabihty of simultaneous occurrence of all three qualities =
= 0.05 x 0.02 x 0.01 = 0.00001
Example 14 :
A university has to select an examiner from a list of 50 persons, 20 of them women and 30 men, 10 of them knowing Hindi and 40 not. 15 of them being teachers and the remaining 35 not. What is the probability of the University selecting a Hindi-knowing women teacher ?
Probability of selecting a women = 20/50
Probability of selecting a teacher =
Probability of selecting a Hindi-knowing candidate =
Since the events are independent the probabihty of the University selecting a Hindi-knowing woman teacher is :
Example 15 :
A ball is drawn at random from a box containing 6 red balls, 4 white balls and 5 blue balls. Determine the probability that it is :
(i) Red (ii) white, (iii) Blue, (iv) Not Red and (v) Red or White.
(i) Probability of Red 6/15 = or 0.40
(ii) Probability of white 4/15 = or 0.267
(iii) Probability of Blue 5/15 = A or 0.333
(iv) Probability of not Red 9/15 = or 0.60
(v) Probability of Red and White 10/15 = or 0.667