91Ó°ÊÓ

A salesman has scheduled two appointments to sell encyclopedias. His first appointment will lead to a sale with probability .$3$, and his second will lead independently to a sale with probability $.6$. Any sale made is equally likely to be either for the deluxe model, which costs $\$1000$, or the standard model, which costs $500

Given,

$P($sale $âˆ\pounds$first $)=0.3$

$P($sale$âˆ\pounds$second $)=0.6$

$P($deluxe $âˆ\pounds$sale $)=0.5$

$P($standard $âˆ\pounds$sale $)=0.5$

$X$is the total dollar value of all sales, which is either $0,500,1000,1500$, or $2000$(because the two sales are worth either $0,500$or $1000$).

$P(X=0)=P($no sale $âˆ\pounds$ first $)×P($ no sale $âˆ\pounds$ second$)=(1-0.3)×(1-0.6)=0.7×0.4=0.28$

$P(X=500)=P($no sale $âˆ\pounds$first $)×P($sale $âˆ\pounds$second $)×P($standard $âˆ\pounds$sale $)$

$+P($sale $âˆ\pounds$first $)×P($no sale $âˆ\pounds$second$)×P($standard $âˆ\pounds$sale $)$

$=(1-0.3)×0.6×0.5+0.3×(1-0.6)×0.5=0.27$

$P(X=1000)=P($sale$âˆ\pounds$first $)×P($sale $âˆ\pounds$second $)×P($standard $âˆ\pounds$sale $)×P($standard $âˆ\pounds$sale $)$

$+P($sale $âˆ\pounds$first $)×P($no sale $âˆ\pounds$second $)×P($deluxe$âˆ\pounds$sale $)$

$+P($no sale $âˆ\pounds$first $)×P($sale$âˆ\pounds$second $)×P($deluxe$âˆ\pounds$sale$)$

$=0.3×0.6×0.5×0.5+0.3×(1-0.6)×0.5+(1-0.3)×0.6×0.5=0.315$

$P(X=1500)=P($sale $âˆ\pounds$first $)×P($sale $âˆ\pounds$second $)×P($standard$âˆ\pounds$sale $)×P($deluxe$âˆ\pounds$sale$)$

$+P($sale $âˆ\pounds$first $)×P($sale $âˆ\pounds$second $)×P($deluxe $âˆ\pounds$sale $)×P($standard $âˆ\pounds$sale $)$

$=0.3×0.6×0.5×0.5+0.3×0.6×0.5×0.5=0.09$

$P(X=2000)=P($sale $âˆ\pounds$first $)×P($sale $âˆ\pounds$second $)×P($deluxe$âˆ\pounds$sale $)×P($deluxe $âˆ\pounds$sale$)$

$=0.3×0.6×0.5×0.5=0.045$

$P(X=0)=0.28$

$P(X=500)=0.27$

$P(X=1000)=0.315$

$P(X=1500)=0.09$

$P(X=2000)=0.045$