Question

Say Your friend is trying to convince you to go out on a blind date with another friend. You have heard them try to convince other people before and know a bit about the probabilities of how they describe dates. When your friend knows it is a great date, they always say they recommend the date. When your friend thinks it is an OK date, your friend recommends the date 75% of the time. When your friend thinks it will be a bad date, they still recommend the date 50% of the time. Generally, 60% of the total amount of dates your friend tries to hookup are clearly bad, while great and OK dates comprise 20% each.

a)​ What is the total probability that your friend recommends the date?
b) ​If your friend recommends the date, what is the probability the date will actually be great?

Answer 1

Solution

Back-up Theory

If A and B are two events such that probability of B is influenced by occurrence or otherwise

of A, then Conditional Probability of B given A, denoted by P(B/A) = P(B ∩ A)/P(A)…...................................…....….(1)

P(B) = {P(B/A) x P(A)} + {P(B/A^C) x P(A^C)}…………………………………………...............................................…….(2)

If A is made up of k mutually and collectively exhaustive sub-events, A1, A2,..Ak,

P(B) = sum over i = 1 to k of {P(B/Ai) x P(Ai)} ………......................................…………………......…..........………....(3)

Baye’s Theorem: P(A/B) = P(B/A) x {P(A)/P(B)}…………………..................…………..…………......….…....…..…….(4)

Now to work out the solution,

Let

A1 represent the event that the date is great,

A2 represent the event that the date is OK,

A3 represent the event that the date is bad

B represent the event that the friend recommends the date.

Then,

‘Generally, 60% of the total amount of dates your friend tries to hookup are clearly bad, while great and OK dates comprise 20% each.’ => P(A1) = 0.2 i.e., 20% , P(A2) = 0.2 i.e., 20% and P(A3) = 0.6 i.e., 60% ............................ (5)

‘When your friend knows it is a great date, they always say they recommend the date.

When your friend thinks it is an OK date, your friend recommends the date 75% of the time.

When your friend thinks it will be a bad date, they still recommend the date 50% of the time.’ =>

P(B/A1) = 1 i.e., always , P(B/A2) = 0.75 i.e., 75% and P(B/A3) = 0.5 i.e., 50% .........................................................(6)

Part (a)

Total probability that the friend recommends the date

= P(B)

= sum over i = 1 to k of {P(B/Ai) x P(Ai)} [vide (3)]

= (1 x 0.2) + (0.75 x 0.2) + (0.5 x 0.6) [vide (5) and (6)]

= 0.2 + 0.15 + 0.3

= 0.65 Answer 1

Part (a)

​If the friend recommends the date, the probability the date will actually be great

= P(A1/B)

= {P(B/A1) x P(A1)}/P(B) [vide (4)]

= (1 x 0.2)/0.65 [vide (6), (5) and Answer 1]

= 0.3077 Answer 2

DONE