A telemarketing company calls telephone numbers chosen at random. It finds that

Question

A telemarketing company calls telephone numbers chosen at random. It finds that 65% of successful calls result in talking to a woman, and that 35% result in talking to a man. After that point, 30% of the women and 20% of the men actually buy something. Let W denote that a woman answered the call. Let Wc denote that a man answered the call. Let S denote that a sale was made. Let Sc denote that a sale was not made. Use the idea of probability tree diagram to find the probabilities of all four "and" events: P(W and S), P(W and Sc), P(Wc and S), P(Wc and Sc). i. What percentage of calls result in a sale? ii. What percentage of sales are made to women? (Write this as a conditional probability.)

Explanation / Answer

P(W and S) =P(Women)*P(Sale|Women)=0.65*0.30=0.195

P(W and Sc)=P(Women)*P(no Sale|Women)=0.65*(1-0.30)=0.455

P(Wc and S) =P(men)*P(Sale|men)=0.35*0.20=0.07

P(Wc and Sc)=P(men)*P(no Sale|men)=0.35*0.8=0.28

i)percentage of calls result in a sale =P(W and S)+P(W and Sc)=0.195+0.07=0.265

ii) percentage of sales are made to women =P(S|W)=P(W and S)/P(S)=0.195/0.265=0.7358 ~ 73.58%

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