opinaj

2022-09-29

"In a recent survey it was found that 50% of investors placed their money in Apple shares while 60% invested in Google. Of the people surveyed, 15% invested in neither of these.
Find the proportion of investors that had both Apple and Google shares."
I have tried calculating as two independent events:
$Pr\left(A\cap B\right)=Pr\left(A\right)\cdot Pr\left(B\right)=0.5\cdot 0.6=0.3,$
which is incorrect.
According to the answers, the correct answer should be 0.25, but I'm not able to find this at all. Any help is greatly appreciated!

### Answer & Explanation

Sanaa Hudson

$P\left(A\right)=0.5,P\left(G\right)=0.6$
$1-P\left(A\cup G\right)=0.15$
$P\left(A\cup G\right)=0.85$
$P\left(A\right)+P\left(G\right)-P\left(A\cap G\right)=0.85$
$0.5+0.6-P\left(A\cap G\right)=0.85$
$P\left(A\cap G\right)=1.1-0.85=0.25$
The reason your method doesn't work is because the event of buying apple stocks and google stocks are not independent.

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