Suppose a traditional painter and a contemporary painter are each asked separately to create artwork for the lobby of a company. Assume one piece is created by each. The probability that the traditional painter is commissioned for more work is 2/3 while there is a 2/5 probability that the contemporary painter is commissioned. There is a 3/4 probability that at least one of them will be commissioned.

a. What is the probability that both are commissioned?

b. What is the probability that the traditional painter was the one commissioned given that only one of them was commissioned?

c. What is the probability that neither is commissioned?

a. 19/60

b. 21/26

c. ¼

Step-by-step explanation:

This problem can be solved using the sets language / notations and rules of probability.

a.

Let probability of traditional painter be written as P (TP) and probability of contemporary painter be written as P (CP)

From the given dа ta:

P (TP) = 2/3

P (CP) = 2/5

P (TP U CP) = ¾ where U represent union of events TP and CP which means at least one of them is commissioned.

The probability of both TP and CP are commissioned can be written as P (TP ∩ CP) where ∩ represents intersection of the two events TP and CP.

Using the rule of probability:

P (AUB) = P (A) + P (B) - P (A∩B)

P (TP U CP) = P (TP) + P (CP) - P (TP ∩ CP)

Substituting the values:

¾ = 2/3 + 2/5 + P (TP ∩ CP)

Rearranging and making P (TP ∩ CP) the subject in the above equation

P (TP ∩ CP) = ¾ - (2/3 + 2/5)

P (TP ∩ CP) = 19/60 (Answer)

b.

It is the question of conditional probability

The rule for conditional probability is that the probability of A conditioned on B, denoted P (A|B), is equal to P (AB) / P (B).

In our case it can be written as P (TP | only one is commissioned) = P (only one is commissioned and TP is the one commissioned) / P (only one is commissioned).

P (only one is commissioned and TP is the one commissioned) = P (TP) - P (both are commissioned)

P (only one is commissioned and TP is the one commissioned) = 2/3 - 19/60

P (only one is commissioned and TP is the one commissioned) = 7/20

Similarly:

P (only one is commissioned and CP is the one commissioned) = 2/5 - 19/60

P (only one is commissioned and CP is the one commissioned) = 1/12

P (only one is commissioned) = P (only one is commissioned and TP is the one commissioned) + P (only one is commissioned and CP is the one commissioned)

P (only one is commissioned) = 7/20 + 1/12

P (only one is commissioned) = 13/30

Therefore:

P (TP | only one is commissioned) = P (only one is commissioned and TP is the one commissioned) / P (only one is commissioned).

P (TP | only one is commissioned) = (7/20) / 13/30

P (TP | only one is commissioned) = 21/26 (Answer)

c.

Using complement rule of probability which is:

P (Neither is commissioned) = 1 - P (at least one is commissioned)

P (TP U CP) ' = 1 - P (TP U CP)

P (TP U CP) ' = 1 - ¾

P (TP U CP) = ¼ (Answer)