Conditional Probability Conditional Probability PA B means the

Conditional Probability

Conditional Probability P(A B) means “the probability that event A occurs given that B has occurred”.

Conditional Probability Ex 1. The following table gives data on the type of car, grouped by gas consumption, owned by 100 people. Low Medium High Male 12 33 7 Female 23 21 4 Total 100 One person is selected at random. L is the event “the person owns a low rated car” F is the event “a female is chosen”. Find (i) P(L) (ii) P(F ∩ L) (iii) P(F L)

Conditional Probability Solution: Male Female Find (i) P(L) = Low 12 23 35 (ii) P(F ∩ L) Medium 33 21 High 7 4 Total 100 (iii) P(F L) (leave the answer as fraction

Conditional Probability Solution: Male Female Low 12 23 Medium 33 21 High 7 4 Total 100 Find (i) P(L) (ii) P(F and L) (iii) P(F L) (i) P(L) = (ii) P(F ∩ L) = The probability of selecting a female with a low rated car.

Conditional Probability Solution: Male Female Find (i) P(L) Low 12 23 35 (ii) P(F and L) Medium 33 21 High 7 4 Total 100 (iii) P(F L) (i) P(L) = (ii) P(F ∩ L) = (iii) P(F L) = The probability of selecting a female given the car is low rated.

Conditional Probability Solution: Male Female Low 12 23 Medium 33 21 High 7 4 Total 100 Find (i) P(L) = (ii) P(F ∩ L) = (iii) P(F L) = (ii) P(F and L) (iii) P(F L)

Conditional Probability • If you don’t have a table, you can use this formula

Conditional Probability Researchers asked people who exercise regularly whether they jog or walk. Fifty-eight percent of the respondents were male. Twenty percent of all respondents were males who said they jog. Find the probability that a male respondent jogs. P( male ) = 58% P( male and jogs ) = 20% Let A = male. Let B = jogs. Write: P( A | B ) = = 0. 2 0. 58 P( A and B ) P( A ) Substitute 0. 2 for P(A and B) and 0. 58 for P(A). 0. 344 Simplify. The probability that a male respondent jogs is about 34%.

Conditional Probability Using Tree Diagrams Jim created the tree diagram after examining years of weather observations in his hometown. The diagram shows the probability of whether a day will begin clear or cloudy, and then the probability of rain on days that begin clear and cloudy. a. Find the probability that a day will start out clear, and then will rain. The path containing clear and rain represents days that start out clear and then will rain. P(clear and rain) = P(rain | clear) • P(clear) = 0. 04 • 0. 28 = 0. 011 The probability that a day will start out clear and then rain is about 1%.

Conditional Probability (continued) b. Find the probability that it will not rain on any given day. The paths containing clear and no rain and cloudy and no rain both represent a day when it will not rain. Find the probability for both paths and add them. P(clear and no rain) + P(cloudy and no rain) = P(clear) • P(no rain | clear) + P(cloudy) • P(no rain | cloudy) = 0. 28(. 96) +. 72(. 69) = 0. 7656 The probability that it will not rain on any given day is about 77%.

Conditional Probability Ex 3 In November, the probability of a man getting to work on time if there is fog on 285 is If the visibility is good, the probability is . . The probability of fog at the time he travels is . (a) Calculate the probability of him arriving on time. (b) Calculate the probability that there was fog given that he arrives on time.

Conditional Probability P(T F) P(T F/) F Fog / FNo Fog P(F) On T time Not on / T time On T time Not on T/ time

Conditional Probability P(T F) P(T F/) P(F) T F T/ F/ T T/ Because we only reach the 2 nd set of branches after the 1 st set has occurred, the 2 nd set must represent conditional probabilities.

Conditional Probability (a) Calculate the probability of him arriving on time. T F T/ F/ T T/

Conditional Probability (a) Calculate the probability of him arriving on time. T F F/ ( foggy and he / T arrives on time ) T T/

Conditional Probability (a) Calculate the probability of him arriving on time. T F T/ F/ T ( not foggy and he T/ arrives on time )

Conditional Probability (b) Calculate the probability that there was fog given that he arrives on time. We need Fog on 285 Getting to work T F From part (a),

Example Conditional Probability A sample of 100 adults were asked how they travelled to work. The results are shown in the table. Walk Bus Bike Car Men 10 12 6 26 Women 7 18 4 17 Total 100 One person is picked at random. M is the event “the person is a man” C is the event “the person travels by car” Find (i) P(M) (ii) P(M C) (iii) P(M/ and C) (iv) P(C M/)

Conditional Probability Solution: Walk Bus Bike Car Men 10 12 6 26 Women 7 18 4 17 Total Find (i) P(M) 54 100 (ii) P(M C) (i) P(M) = (iii) P(M/ and C) (iv) P(C M/) (ii) P(M C) =

Conditional Probability Solution: Walk Bus Bike Car Men 10 12 6 26 Women 7 18 4 17 Total Find (i) P(M) 43 (ii) P(M C) (i) P(M) = (iii) P(M/ and C) 54 100 (iv) P(C M/) (ii) P(M C) =

Conditional Probability Solution: Walk Bus Bike Car Men 10 12 6 26 Women 7 18 4 17 Total Find (i) P(M) 43 (ii) P(M C) (iii) P(M/ and C) 54 100 (iv) P(C M/) (i) P(M) = (ii) P(M C) = (iii) P(M/ and C) = (iv) P(C M/) =

Conditional Probability Solution: Walk Bus Bike Car Men 10 12 6 26 54 Women 7 18 4 17 46 43 100 Total Find (i) P(M) (ii) P(M C) (iii) P(M/ and C) (iv) P(C M/) (i) P(M) = (ii) P(M C) = (iii) P(M/ and C) = (iv) P(C M/) =

Conditional Probability Independent Events If the probability of the occurrence of event A is the same regardless of whether or not an outcome B occurs, then the outcomes A and B are said to be independent of one another. Symbolically, if then A and B are independent events.