Suppose the events Upper B 1, Upper B 2, and Upper B 3, are mutually exclusive and complementary events, such that P(Upper B 1)equals0.05, P(Upper B 2)equals0.4, and P(Upper B 3)equals0.55. Consider another event A such that P(A|Upper B 1)equals0.6, P(A|Upper B 2)equals0.5, and P(A|Upper B 3)equals0.4. Use Bayes's Rule to find P(Upper B 1|A).

0.0666

Step-by-step explanation:

We have three events that we are said are mutually exclusive and complementary. That is, the union of the three events is the sample space. This is confirmed by checking that the probabilities of B1, B2 and B3 add up to 1. P(B1) = 0.05, P(B2)=0.4 and P(B3) = 0.55 do add up to 1.

Consider the definition of conditional probability. Given two events A,B such that P(B)>0 we have that

Note that this implies that

We have that,

Note that from that expression, we are only missing P(A). REcall that since B1,B2 and B3 union up to the sample space, we have that

Hence, since this three sets are disjoint, we have that

Using the previous relation that we had, we have that

Then, the final answer is

14

step-by-step explanation:

14

let x and 400 - x be the two numbers.  

==> 0.80x + 0.85(400 - x) = 400 - 68 from the statement of the problem.  

< ==> 0.80x + 0.85(400 - x) = 400 - 68  

==> 340 - 0.05x = 332  

< ==> -0.05x = -8 ==> x = 160, the first number ==> 400 - x = 240, the second number.