# Difference between revisions of "Eleisha's Segment 2: Bayes"

To Calculate:

1. If the knight had captured a Gnome instead of a Troll, what would his chances be of crossing safely?

TTGGG bridges (20%)

TGGGG bridges (20%)

GGGGG bridges (60%)

If the knight captures a gnome, the only possibility of a safe crossing if he crosses one of the third type of bridge (GGGGG).

Hypothesis 1 (H1) = Knight crosses a TTGGG bridge

Hypothesis 2 (H2) = Knight crosses a TGGGG bridge

Hypothesis 3 (H3) = Knight crosses a GGGGG bridge $\displaystyle P(H_2 | G) = \frac{P(G| H_2)P(H_2)}{P(G | H_1)p(H_1) + P(G| H_2)P(H_2) + P(G | H_3)P(H_3) } =\frac{(1)(\frac{3}{5})}{(\frac{3}{5})(\frac{1}{5}) + (\frac{4}{5})(\frac{1}{5}) + (1)(\frac{3}{5})} \approx 0.682$

2. Suppose that we have two identical boxes, A and B. A contains 5 red balls and 3 blue balls. B contains 2 red balls and 4 blue balls. A box is selected at random and exactly one ball is drawn from the box. What is the probability that it is blue? If it is blue, what is the probability that it came from box B?

A = Drawn from Box A

B = Drawn from Box B

Probability that the drawn ball is blue = $\displaystyle P(Blue) = P(Blue | A)P(A) + P(Blue | B)P(B) = \left(\frac{1}{2}\right)\left(\frac{3}{8}\right) + \left(\frac{1}{2}\right)\left(\frac{4}{6}\right) = \frac{25}{48} \approx 0.521$

Using Bayes and the observation that the ball drawn was Blue, we can can calculate the probability that the ball is from box B

Hypothesis 1 (H1) = The ball was drawn from box A

Hypothesis 2 (H2) = The ball was drawn from box B

Probability that if the drawn ball is blue then it came from box B = $\displaystyle P(H_2 | Blue)$ $\displaystyle P(H_2 | Blue) = \frac{P(Blue | H_2)P(H_2)}{P(Blue | H_1)p(H_1) + P(Blue| H_2)P(H_2)} = \frac{(\frac{1}{2})(\frac{4}{6})}{ (\frac{3}{16}) + (\frac{4}{12})} = \frac{16}{25} = 0.64$