Question

You are about to make an offer on a new pair of snowboard boots you found in a craigslist ad. The ad says the previous owner broke his wrist snowboarding and can no longer use the boots, so it seems like you can get a good deal (since they now have no value to the owner). The only problem is you don’t knowhowvaluetheyhavetootherpeoplewhoarealsoviewingthead. Youwanttoofferaslittleas possible without being outbid by another party. The probability that the boot owner has a higher offer, denotedbyH,isassumedtobe1⁄4. Ifthebootownerhasahigheroffer,he/sheisunlikelytoaccept youroffertobuytheboots. Infact,youthinktheprobabilityisonly1/6thatthebootownerwillsellto you if they have a higher offer (perhaps because dealing with you is more convenient). However, if the boot owner does not have a higher offer, you are pretty sure they will accept your bid. Of this, you estimate there is 2/3 probability that your offer will be accepted. Based upon this information:

1. a) What is the probability that the boot owner has a higher offer but will accept your offer anyway?

2. b) What is the probability that the boot owner will accept your offer to buy the boots?

3. c) Suppose the boot owner accepts your offer to buy the boots. What's the probability that they had a

   higher offer from another party, given that they agreed to sell you the boots?

Answer 1

a) Let H denote the event that boot owner has a higher offer

P(H) =1/4

Let A denote the event that boot owner accept your offer

P( AI H) = 1/6

To find P( A H) = ?

We know that ,

P( A H) = P( A I H) *P(H)

= (1/6) *(1/4)

= 0.0417

(b)

Let H denote the event that boot owner has a higher offer

and H' is the event that boot owner doesnot have a higher offer

P(H) =1/4

P( H') = 3/4

Let A denote the event that boot owner accept your offer

P( AI H) = 1/6

Also P( A I H') = 2/3

To find P(A)

We know that ,

P(A) = P( AI H) *P(H) + P( A I H') *P(H')

= (1/6) *(1/4) + (2/3) *(3/4)

= 0.5417

(c) To find

P( H I A) = ?

Using Bayes' theorem

P( H I A) = P( A I H) *P( H) / P(A)

= 0.0417 / 0.5417

= 0.0770