Addition Rule of Probability |
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| Suppose we throw a dice. What is the probability of an old number or a prime number coming up on the dice? |
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If A is the event of getting an odd number and
B is the event of getting a prime number, |
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Then, Outcomes for A = 3 {1, 3, 5}
Outcomes for B = 3 {2, 3, 5} |
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| A∪B = A or B = getting an odd number or a prime number = {1, 2, 3, 5} |
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| A∩B = A and B = getting an odd prime number = {3, 5} |
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| Thus P (A ∪ B) = P (A) + P (B) – P(A ∩ B) |
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| This result is called the Addition Rule of Probability. |
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If however the events are mutually exclusive then P(A∩B)= 0
and P(A∪B) = P(A) + P(B) |
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Multiplication rule of probability |
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| Suppose a coin is tossed twice. |
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Let A be the event of 'getting a head in the first toss’
B the event of 'getting a tail in the 2nd toss' |
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| Sample space = {HH, HT, TH, TT} = 4 = Total number of outcomes |
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| A = {HH, HT} = 2 B = {HT, TT} = 2 |
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| (A∩B) = {HT} =1 |
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| The above events are independent events. If the events are dependent then |
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| P (A∩B) # P (A) * P (B) |
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| Example : 22 |
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| 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? |
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| Solution: |
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| When two dice are cast the total number of outcomes = 6 * 6 = 36 |
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Sample space S = |
{(1, 1), |
(1, 2), |
(1, 3), |
(1, 4), |
(1, 5), |
(1, 6) |
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(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)} |
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Number of outcomes favorable to E = 2 {(5, 6) (6, 5)} |
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| Number of outcomes favorable to F = 30 |
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| Since we exclude the set {(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6)} |
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| Thus, E and F are dependent events |
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| In the above problem the pairs (1, 1) (2, 2) (3, 3) (4, 4) (5, 5) (6, 6) are called doublets. |
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Try these problems |
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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 |
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| Answer: |
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Total number of outcomes = Total number of cards
n(S) = 52 |
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