Suppose the events B1, B2, and B3 are mutually exclusive and complementary events, such that P(B1) = .2, P(B2) = .15, and P(B3) = .65. Consider another event A such that P(A|B1) = .4, P(A|B2) = .25, and P(A|B3) = .6. Use Bayes’s Rule to find

a. P(B1|A)

b. P(B2 |A)

c. P(B3 |A)

Step 1 of 3:

Let the events , , and are mutually exclusive and complementary events.

So , , and .

We consider another , , and .

Our goal is:

a). We need to find .

b). We need to find .

c). We need to find .

a).

Now we use Bayes’s rule we have to find .

First we need to find .

=

We substitute and values.

0.8

Therefore, 0.8.

Now we need to find .

=

We substitute and values.

0.0375

Therefore, 0.0375.

Then we need to find .

=

We substitute and values.

0.39

Therefore, 0.39.

Now we have to find P(A).

P(A) = ++

We know that , , and .

P(A) = 0.8+0.0375+0.39

P(A) = 0.5075

Therefore, P(A) = 0.5075.

Now we use Bayes’s rule we have to find .

The formula for is

=

We know that and .

=

= 0.642

Therefore, = 0.642.

Step 2 of 3:

b).

Now we use Bayes’s rule we have to find .

The formula for is

=

We know that and .

=

= 0.073

Therefore, = 0.73.