Q.3.35 - A First Course in Probability

Question

With probability $.6$, the present was hidden by mom; with probability $.4$, it was hidden by dad. When mom hides the present, she hides it upstairs $70$ percent of the time and downstairs $30$ percent of the time. Dad is equally likely to hide it upstairs or downstairs.

(a) What is the probability that the present is upstairs?

(b) Given that it is downstairs, what is the probability it was hidden by dad?

Step-by-Step Solution

Answer

(a) The probability that the present is upstairs is $0.62$.

(b) The probability it was hidden by dad if the given are down stairs is $0.5263$.

Step 1: Given information (Part a)

With probability $.6$, the present was hidden by mom; with probability $.4$, it was hidden by dad. When mom hides the present, she hides it upstairs $70$ percent of the time and downstairs $30$ percent of the time. Dad is equally likely to hide it upstairs or downstairs.

We need to find that  the probability that the present is upstairs . 

Step 2: Solution (Part a)

The solution is,

$A=$event that present was hidden by mom

$B=$event that present was hidden by dad.

$E=$event that she hides in upstairs.

$F=$event that she hides in downstairs.

Then,

$P\left(A\right)=.6$

$P\left(B\right)=.4$

$P(E\mid A)=.70$

$P(F\mid A)=.30$

$P(E\mid B)=.5$

$P(F\mid B)=.5$

Step 3: Final solution (Part a)

The probability that the present is upstairs is,

$P\left(E\right)=P\left(A\right)P(E\mid A)+P\left(B\right)P(E\mid B)$

$=\left(.6\right)\left(.70\right)+\left(.4\right)\left(.5\right)$

$=.42+.20$

$=.62$

Step 4: Final answer (Part a)

The probability that the present is upstairs is $0.62$. 

Step 5: Given information (Part b)

With probability $.6$, the present was hidden by mom; with probability $.4$, it was hidden by dad. When mom hides the present, she hides it upstairs $70$ percent of the time and downstairs $30$ percent of the time. Dad is equally likely to hide it upstairs or downstairs.

We need to find that is the probability it was hidden by dad 

Step 6: Solution (Part b)

The probability that it was hidden by dad if it is downstairs will be,

$P(B\mid F)=\frac{P(F\mid B)P\left(B\right)}{P(F\mid A)P\left(A\right)+P(F\mid B)P\left(B\right)}$

$=\frac{\left(.5\right)\left(.4\right)}{\left(.30\right)\left(.6\right)+\left(.5\right)\left(.4\right)}$

$=\frac{.20}{.38}$

$=\frac{10}{19}$

$\approx 0.5263$

Step 7: Final answer (part b)

The probability it was hidden by dad if the given are down stairs is $0.5263$.