400 Q2: Consider the data given which is related to the results of a knee operat

Question

400 Q2: Consider the data given which is related to the results of a knee operation at a medical department. Success Fast Success Slow Unsuccess Young Light 1000 150 50 Young Heavy 500 300 100 Old Light 400 200 Old Heavy 200 600 300 Answer the following: (a) Define the Sample space, Events, and their corresponding probabilities. (b) Check that the obtained probabilities represent a pmf. (c) What is the probability that an Old man will have a successful surgery with a speedy recovery. (d) If a patient undegoes an operation, what is the probability that the result will be unsuc- cessful. e) Assuming that an operation was successful, what is the probability that the patient was: i. Young; ii. Old; iii. Comment (i.e., do you expect these results).

Explanation / Answer

Consider the events

A: Type of patients and B : Recovery of operation

The sample space is given by :

S=(w1,w2,..........,w12)

where wi (i=1:12) be the outcomes of the sample space.

where w1: Patient is young light and recovery of operation is sucessfully fast.

w2: Patient is young light and recovery of operation is sucessfully slow

w3: Patient is young light and operation is not sucessful.

w4: Patient is young heavy and recovery of operation is sucessfully fast.

w5: Patient is young heavy and recovery of operation is sucessfully slow

w6: Patient is young heavy and operation is not sucessful.

w7: Patient is old light and recovery of operation is sucessfully fast.

w8: Patient is old light and recovery of operation is sucessfully slow

w9: Patient is old light and operation is not sucessful.

w10: Patient is old heavy and recovery of operation is sucessfully fast.

w11: Patient is old heavy and recovery of operation is sucessfully slow

w12: Patient is old heavy and operation is not sucessful.

Probabilities of outcome of sample space are

P(w1) =1000/4200 P(w2) =150/4200 =P(w3) =50/4200

P(w4) =500/4200 P(w5) =300/4200 P(w6) =100/4200

P(w7) =400/4200 P(w8) =600/4200 P(w9) =100/4200

P(w10) =200/4200P(w11) =600/4200 P(w12) =300/4200

The Probability distribution of events A is

Since total probabilty is one . hence prob. distribution of A represents p.m.f

The probability distribution of event B is

Type of

recovery

Since total probabilty is one . hence prob. distribution of B represents p.m.f

Joint probability distribution of A & B

c) P(Old man will have succesful surgery with speedy recovery)

=P(old light and Success fast) +P(old heavy and sucess fast)

= 400/4200+200/4200=600/4200 =1/7

= 0.1428

d) P( Operation will be unsuccessful) = 650/4200= 0.1547

e) i) P( operation was successful and patient was young)

= P( young light and Sucess fast)+P(young heavy and sucess fast) +  P( young light and Sucessslow)+P(young heavy and sucess slow)

= 1000/4200+500/4200+150/4200+300/4200= 1950/4200

=0.4642

ii) P( operation was successful and patient was old)

= P(old light and Sucess fast)+P(old heavy and sucess fast) +  P(old light and Sucessslow)+P(old heavy and sucess slow)

= 400/4200+200/4200+400/4200+600/4200= 1600/4200

= 0.3809

iii) The propertion of successful operation of young person is greater than old person.

Patient Young light Young heavy old light old heavy total Probability 1200/4200 900/4200 1000/4200 1100/4200 1

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