Question

You have a dentist appointment for teeth filling. During your visit your regular doctor is not available but instead you are greeted by Dr. Sweeney Todd. Dr. Todd performs a tooth surgery on you but his skills in identifying which region to fill are not very accurate. He decides that he will fill the top right molar (TR) with probability 0.5, Top left molar with probability (TL) 0.6, and bottom left molar with probability (BL) 0.1. In addition, to the lack of certainty Dr. Todd takes 3 tries to fill the region identified earlier. Suppose that the event corresponding to successful surgery for a given region on a single trial, is denoted by S and if the surgery is unsuccessful, the event is denoted by N for a single region.

(a) What is P(NNS)? (That is, it is not successful for the first 2 tries and is successful for the last try)

(b) Assuming that the three trials, in order, are NNS, what is the probability that the tooth region Dr. Todd picked for filling was the top right molar (TR)?

Answer 1

E_1 be the event Dr. Todd performs a tooth surgery on top right molar (TR)

P[ Dr. Todd performs a tooth surgery on top right molar (TR) ] = P[ E_1 ] = 0.5

E_2 be the event Dr. Todd performs a tooth surgery on Top left molar (TL)

P[ Dr. Todd performs a tooth surgery on Top left molar (TL) ] = P[ E_2 ] = 0.6

E_3 be the event Dr. Todd performs a tooth surgery on bottom left molar (BL)

P[ Dr. Todd performs a tooth surgery on bottom left molar (BL) ] = P[ E_3 ] = 0.1

a) NNS be the event it is not successful for the first 2 tries and is successful for the last try

P[ NNS | E_1 ] = 0.5*0.5*0.5 = 0.125 ( two failures and last one success, assuming equal probability of success and failure )

P[ NNS | E_2 ] = 0.5*0.5*0.5 = 0.125 ( two failures and last one success, assuming equal probability of success and failure )

P[ NNS | E_3 ] = 0.5*0.5*0.5 = 0.125 ( two failures and last one success, assuming equal probability of success and failure  )

P[ NNS ] = P[ NNS | E_1 ]*P[ E_1 ] +  P[ NNS | E_2 ]*P[ E_2 ] +  P[ NNS | E_3 ]*P[ E_3 ]

P[ NNS ] = 0.5*0.125 + 0.6*0.125 + 0.1*0.125 = 0.15

P[ NNS ] = 0.15

b) Probability that the tooth region Dr. Todd picked for filling was the top right molar (TR) = P[ E_1 | NNS ]

P[ E_1 | A ] = P[ NNS | E_1 ]*P[ E_1 ] / P[ NNS ] = 0.5*0.125 / 0.15 = 0.4167

P[ E_1 | A ] = 0.4167