125 views

Doctors $A$ and $B$ perform surgery on patients in stages $III$ and $IV$ of a disease. Doctor $A$ has performed a $100$ surgeries (on $80$ stage $III$ and $20$ stage $IV$ patients) and $80$ out of her $100$ patients have survived ($78$ stage $III$ and $2$ stage $IV$ survivors). Doctor $B$ has also performed $100$ surgeries (on $50$ stage $III$ and $50$ stage $IV$ patients). Her success rate is $60/100$ ($49$ stage $III$ survivors and $11$ stage $IV$ survivors).$A$ patient has been advised that she is equally likely to be suffering from stage $III$ or stage $IV$ of this disease. Which doctor would you recommend to this patient and why?

1. Doctor $A$ since she has a higher success rate
2. Doctor $A$ since she specializes in stage III patients and the success of surgery in stage $IV$ patients is anyway too low
3. Doctor $B$ since she has performed more stage $IV$ surgeries
4. Doctor $B$ since she appears to be more successful
5. There is not enough data since the choice depends on the stage of the disease the patient is suffering from.

Ans will be d

As, %of Doctor A successful for stage III 78/80*100=97.5

for stage IV 2/20*100=10

As, %of Doctor B successful for stage III 49/50*100=98

stage IV 11/50*100=22

In both cases Doc B is more successful

selected by
Stage 4 is much harder to treat than stage 3 is implied in question.

probabilistic approach

Probability that the patient will survive if operated by A = 1/2*78/80 + 1/2*2/20 = 86/160

Probability that the patient will survive if operated by B = 1/2*49/50 + 1/2*11/50 = 60/100 which is greater

This question is based on Simpson's Paradox.

1.Success rate of Dr.B in Stage Ⅲ (49/50 =98%)Success rate of Dr.A in Stage Ⅲ(78/80 = 97.5%)

2.Success rate of Dr.B in Stage Ⅳ (11/50 = 22%) > Success rate of Dr.A in Stage  Ⅳ(2/20 =10%)

3.The aggregate Success rate of Dr.B (60%)The aggregate Success rate of Dr.A (80%)

I recommend Dr. B because her success rate in both stages Ⅲ and Ⅳ is greater than Dr. A.Individual success rate is matter rather than aggregate.

## Correct ans is (d)Doctor B since she appears to be more successful

answered ago by Boss (5.7k points)