Kendra has programmed her cell phone to randomly show one of six photos when she turns it on. Two of the photos are of her parents, one is of her niece, and three are of her boyfriend, Bruce. Today, she will need to turn her phone on twice: once before school and again after school.

1. Create an area model to represent this situation.

2. Given that the before-school photo was of her boyfriend, what is the conditional probability the after-school photo will also be of her boyfriend?

3. What is the probability that neither photo will be of her niece?

4. Given that neither photo was of her niece, what is the conditional probability that the before-school photo was of her boyfriend?

   Hint (a):

   Create a model that displays ALL possible scenarios.
   Probability tree: 3 branches, labeled as follows: P branch, 2 sixths, N branch, 1 sixth, and B branch, 3 sixths. Each branch, splits into the same 3 branches, again. 3 by 3 rectangle, labeled as follows: Top edge, & left edge each labeled, P, 1 third, N, 1 sixth, B, 1 half. Interior: Top row: 1 ninth, 1 eighteenth, 1 sixth. Middle row: 1 eighteenth, 1 thirty sixth, 1 twelfth. Bottom row: 1 sixth, 1 twelfth, 1 fourth.

   Help (b):

   On probability tree, last branch is circled. On area model, bottom right interior, square with 1 fourth, is circled.

   Circle all the places on both models where both photos are Kendra's boyfriend.

   Answer (b):

   P(both boyfriend) =

   $\frac{\frac{1}{4}}{\frac{1}{2}}=\frac{1}{2}$

   Help (c):

   Circle all the places on both models where neither photo is Kendra's niece.
   On probability tree, the following branches are circled: top & bottom from, P branch, top & bottom from, b, branch. On area model, the following interior squares are circled: top left, 1 ninth, top right, 1 sixth, bottom left, 1 sixth, bottom right. 1 fourth.

   Answer (c) :

   P(not niece) =

   $\frac{4}{36}+\frac{6}{36}+\frac{6}{36}+\frac{9}{36}=\frac{25}{36}\approx0.6944$