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## Addition and Multiplication Rules of Probability

Suppose we throw a dice. What is the probability of an old number or a prime number coming up on the dice?

If          A is the event of getting an odd number and

B is the event of getting a prime number,

Then,   Outcomes for A = 3 {1, 3, 5}
Outcomes for B = 3 {2, 3, 5}

A∪B = A or B = getting an odd number or a prime number = {1, 2, 3, 5}

A∩B = A and B = getting an odd prime number = {3, 5}

Thus P (A ∪ B) = P (A) + P (B) – P(A ∩ B)

This result is called the Addition Rule of Probability.

If however the events are mutually exclusive then P(A∩B)= 0
and P(A∪B) = P(A) + P(B)

### Multiplication rule of probability

Suppose a coin is tossed twice.

Let       A be the event of 'getting a head in the first toss’
B the event of 'getting a tail in the 2nd toss'

Sample space = {HH, HT, TH, TT} = 4 = Total number of outcomes

A = {HH, HT} = 2          B = {HT, TT} = 2

(A∩B) = {HT} =1

The above events are independent events. If the events are dependent then

P (A∩B) # P (A) *  P (B)

Example : 22

Two dice are tossed, if E is the event of 'getting the sum of the numbers on the dice as 11' and F is the event 'getting a number other than 5 on the first dice’, find P (E∩F). Are E and F independent events?

Solution:

When two dice are cast the total number of outcomes = 6 *  6 = 36

 Sample space S = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6) (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6) (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6) (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6) (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6) (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}

Number of outcomes favorable to E = 2 {(5, 6) (6, 5)}

Number of outcomes favorable to F = 30

Since we exclude the set {(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6)}

Thus, E and F are dependent events

In the above problem the pairs (1, 1) (2, 2) (3, 3) (4, 4) (5, 5) (6, 6) are called doublets.

### Try these problems

A card is drawn at random from a well shuffled deck of cards. Find the probability that the card is a

a. King or a red card

b. Club or a diamond

c. Neither a heart nor a king