1
$\begingroup$

A denotes event that present was hidden by mom. B denotes event that present was hidden by dad. E denotes event that present was hidden upstairs. F denotes event that present was hidden downstairs.

P(A)=.6 P(B)=.4 P(E|A)=.7 P(F|A).3 P(E|B)=.5 P(F|B)=.5

Find P(E)

Can you explain why the answer is: P(E) = P(A)P(E|A)+P(B)P(E|B) = (.6)(.7)+(.4)(.5) = .42 +.2 = .62

  • 0
    What do you think the probability is that mom hid the present upstairs? And the probability that dad hid the present upstairs?2012-09-18

1 Answers 1

2

The present could be upstairs in two different (and disjoint) ways: (i) Mom hid the hiding and hid the present upstairs or (ii) Dad did the hiding and hid it upstairs.

What is the probability of (i)? In symbols, it is $\Pr(A\cap E)$. But we have the general formula $$\Pr(X|Y)\Pr(Y)=\Pr(X\cap Y)=\Pr(Y\cap X).$$ Putting $X=E$ and $Y=A$ gives us $\Pr(A\cap E)$.

Do a similar calculation for the probability of (ii), that is, $\Pr(B\cap E)$.

Then add up.

  • 0
    @AndreNicholas Actually the problem did not state the the present could only be hidden in two disjoint ways. But it can be inferred since A and B are clearly disjoint events and we are given P(A)=0.6 and P(B)=0.4. But this will leave the following: P(A$^c$∩B$^c$)=1-P(A)-P(B)=0.2012-09-18
  • 0
    @AndreNicholas can you explain why we need to **ADD** the two to get P(E)?2012-09-18
  • 0
    The event $E$ can happen in two different ways. Or, if you are familiar with "tree" language, it can happen along two different paths. The two ways are "Mom hid it and it is upstairs" and "Dad hid it and it is upstairs." We found the probabilities (i) and (ii) of these. Or else in symbols, $E=(E\cap A) \cup (E\cap B)$. The two events $E\cap A)$ and $E\cap B)$ are *disjoint* so to find probability of union we add.2012-09-18
  • 0
    thank you Andre, i get it now, can you perhaps take a look at this: http://math.stackexchange.com/questions/198697/bayes-theorem-confusion-more-complex it's abit more complex version of the Bayes2012-09-18