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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 $\frac{600}{100}$ ($49$ 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.
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Best answer
14 votes
14 votes

Answer will be  $d)$

As, $\%$ of Doctor $A$ successful for stage III =$\dfrac{78}{80}\times 100=97.5$

For stage IV=$\dfrac{2}{20}\times100=10$

As, $\%$ of Doctor $B$ successful for stage III =$\dfrac{49}{50}\times 100=98$

For stage IV=$\dfrac{11}{50}\times100=22$

In both cases Doc $B$ is more successful. 

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2 votes

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

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0 votes

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

Answer:

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