Events A and B are said to be independent if the occurrence of A does not affect the probability of B.

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1 Independent Events Events A and B are said to be independent if the occurrence of A does not affect the probability of B. Probability experiment of flipping a coin and rolling a dice. Sample Space: {(H, 1) (H, 2), (H, 3), (H, 4), (H, 5), (H, 6), (T, 1), (T, 2), (T, 3), (T, 4), (T, 5), (T, 6)} Let event A = "getting tails on a coin" Let event B = "getting a number less than 3 on a dice" Hence, (A and B) = "getting tails on a coin and a number less than 3 on a dice" = { (T, 1), (T, 2)} P(A) = Probability of getting tails = 1/2 P(B) = Probability of getting a number less than 3 = 2/6 = 1/3 P(A and B) = 2/12 = 1/6

2 The events A and B are called independent events because the occurrence of event A does not affect the probability of event B. Note that what we get when flipping a coin does not affect what number we obtain when tossing a dice. When two events A and B are independent, P(A and B) = P(A) P(B) For the above example, P(A and B) = P(A) P(B) = 2 3 6

3 Presidential General Election Results (National 2008) Party Percentage Democratic 52.87% Republican 45.60% Independent 0.00% Other Source: % If three voters are selected randomly, what is the probability that all three voted for a democratic presidential candidate? (We can assume that the three persons do not know each other.) Answer: We can assume that the outcomes of this probability experiment are independent since in general how one person votes does not affect how another person votes. P(all three voted democratic) = P(1st person voted democratic) P(2nd person voted democratic) P(3rd person voted democratic) 3 =

4 Presidential General Election Results (National 2008) Party Percentage Democratic 52.87% Republican 45.60% Independent 0.00% Other Source: % If ten voters are selected randomly, what is the probability that all ten voted for a republican presidential candidate? (We can assume that the ten persons do not know each other.) Answer: We can assume that the outcomes of this probability experiment are independent since in general how one person votes does not affect how another person votes. P(all ten voted republican) = P(1st person voted republican) P(2nd person voted republican) P(3rd person voted republican) P(4th person voted republican) P(5th person voted republican) P(6th person voted republican) P(7th person voted republican) P(8th person voted republican) P(9th person voted republican) P(10th person voted republican) = =

5 At Least and At Most Probability Probability Experiment: Tossing a coin three times Sample Space = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} Number of outcomes with three tails = 1 Number of outcomes with exactly two tails = 3 Number of outcomes with exactly one tail = 3 Number of outcomes with no tail = 1 Sample Space = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} Set of outcomes with "at least" one tail = {HHT, HTH, HTT, THH, THT, TTH, TTT} Number of outcomes with "at least" one tail = 7 Number of outcomes with no tail = 1 The complement of "at least one tail" is "no tail".

6 Sample Space = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} Set of outcomes with "at least" two tails = {HTT, THT, TTH, TTT} Number of outcomes with at least two tails = 4 Set of outcomes with at most one tail = {HHH, HHT, HTH, THH} Number of outcomes with at most one tail = 4 Note: "at least two tails" is equivalent to "two or more tails". "at most one tail" is equivalent to "less than or equal to 1 tail" Hence, the complement of "at least two tails" is "less than or equal one tail"

7 Probability of Disjoint Events Sample Space = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} Let event E 1 = "getting H on 1st toss AND getting H on 2nd toss AND getting H on 3rd toss" Let event E 2 = "getting H on 1st toss AND getting H on 2nd toss AND getting T on 3rd toss" Let event E 3 = "getting H on 1st toss AND getting T on 2nd toss AND getting H on 3rd toss" Let event E 4 = "getting H on 1st toss AND getting T on 2nd toss AND getting T on 3rd toss" Let event E 5 = "getting T on 1st toss AND getting H on 2nd toss AND getting H on 3rd toss" Let event E 6 = "getting T on 1st toss AND getting H on 2nd toss AND getting T on 3rd toss" Let event E 7 = "getting T on 1st toss AND getting T on 2nd toss AND getting H on 3rd toss" Let event E 8 = "getting T on 1st toss AND getting T on 2nd toss AND getting T on 3rd toss" If we examine all eight events, we see that none of the have anything in common. Hence, all eight events are disjoint (or mutually exclusive). For disjoint events,

8 P(E 1 or E 2 or E 3 or E 4 or E 5 or E 6 or E 7 or E 8 ) = P(E 1 ) + P(E 2 ) + P(E 3 ) + P(E 4 ) + P(E 5 ) + P(E 6 ) + P(E 7 ) + P(E 8 ) In general, for disjoint events E 1, E 2,..., E n, P(E 1 or E 2 or... or E n ) = P(E 1 ) + P(E 2 ) + P(E 3 ) P(E n ) Sample Space = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} What is the probability of getting at exactly H,T,H when a coin is tossed three times? (Assume that the coin is fair.) P(HTH) = P(H) P(T) P(H) =

9 What is the probability of getting at exactly T,T,T when a coin is tossed three times? (Assume that the coin is fair.) P(TTT) = P(T) P(T) P(T) = What is the probability of getting least two tails when a coin is tossed three times? Answer: "at least two tails" means two or more tails. Outcomes with two or more tails are: {HTT, THT, TTH, TTT} P(getting at least two tails) = P(HTT or THT or TTH or TTT) = P(HTT) + P( THT) + P(TTH) + P(TTT) = P(H) P(T) P(T) + P(T) P(H) P(T) + P(T) P(T) P(H) + P(T) P(T) P(T) =

10 What is the probability of getting least one tail when a coin is tossed three times? Answer: "at least one tail" means one or more tails. Outcomes with two or more tails are: {HHT, HTH, HTT, THH, THT, TTH, TTT} P(getting at least two tails) = P(HHT or HTH or HTT or THH or THT or TTH or TTT) = P(HHT) + P(HTH) + P(HTT) + P(THH) + P(THT) + P(TTH) + P(TTT) = Alternative Solution: The complement of "at least one tail" is "no tail". Thus P(at least one tail) = 1 - P(no tail) = 1 - P(HHH) = 1 = 8 8

Mathematics. ( : Focus on free Education) (Chapter 16) (Probability) (Class XI) Exercise 16.2

Mathematics. ( : Focus on free Education) (Chapter 16) (Probability) (Class XI) Exercise 16.2 ( : Focus on free Education) Exercise 16.2 Question 1: A die is rolled. Let E be the event die shows 4 and F be the event die shows even number. Are E and F mutually exclusive? Answer 1: When a die is