Question

Suppose that one box contains 201 long bolts and 323 short bolts ,and that the other box contains 505 long bolts and 412 short bolts. Suppose also that one box is selected at random and a bolt is then selected at random from that box. Find the probability that this bolt is long.

Answer 1

Solution:

First, let us calculate the probability of getting a long bolt from the first box:

$P(l|box_1)= \frac{201}{201+323}=\frac{201}{524}$

Secondly, we will caculate the probability of getting a long bolt from the second box:

$P(l|box_2) = \frac{505}{412+505}=\frac{505}{917}$

Now, teh probability of choosing a box is:

$P(box_1)=P(box_2)=\frac{1}{2}$

And finally, the probability of getting a long bolt:

$P(l) = P(l|box_1)P(box_1) + P(l|box_2)P(box_2) = \frac{1}{2}*\frac{201}{524}+$

$+ \frac{1}{2}*\frac{505}{917} = \frac{1}{2}(\frac{201}{524}+\frac{505}{917})=\frac{1}{2}*\frac{3427}{3668}\approx0.467$

Answer:

$\approx0.467$