David has two boxes containing shapes.
In box A there are 4 stars and 2 hearts.
In box B there are 2 stars and 1 heart.
David takes, at random, a shape from box A and puts it into box B.
He then takes a shape from box B.
What is the probability that this shape is a star?
A. {\frac{1}{12}}
B. {\frac{4}{9}}
C. {\frac{2}{3}}
D. {\frac{3}{4}}
E. {\frac{4}{3}}
What we can do is divide this question into two scenarios, keep in mind they need to end in David picking a star from Box B- so it narrows things down.
The probability David takes a heart out of Box A and inserts it into Box B is 1/3.
The probability he takes a star out of Box A and inserts it into Box B is 2/3.
We can now venture into two possibilities:
Scenario 1:
David takes a heart from Box A and inserts it into Box B.
David then picks a star from Box B.
We can now visualize that the probability he picks a inserts a heart is 1/3 and then picks a star would be 1/2.
We can permute this total scenario to be: \frac{1}{3} \cdot \frac{1}{2} = \frac{1}{6}
Scenario 2:
David takes a star from Box A and inserts it into Box B.
David then picks a star from Box B.
We can now visualize that the probability he picks a inserts a star is 2/3 and then picks a star would be 3/4.
We can permute this total scenario to be: \frac{2}{3} \cdot \frac{3}{4} = \frac{1}{2}
Our final step is to add these probabilities to formulate the possibility of him picking a star in general, which means to add these two scenarios together:
\frac{1}{6} + \frac{1}{2} = \frac{2}{3}
Leaving our answer to be C, 2/3.
