SOLUTION: Table below are listed the numbers of doctors in various specialties by gender. Pathology Pediatrics Psychiatry Male 12575 33020 27803 Female 5604 33351

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Question 1172421: Table below are listed the numbers of doctors in various specialties by gender.
Pathology Pediatrics Psychiatry
Male 12575 33020 27803
Female 5604 33351 12292
Choose one doctor at random and find:
(c) Are the characteristics “female” and “pathologist” independent? Explain clearly

Found 2 solutions by ikleyn, math_tutor2020:
Answer by ikleyn(52781)   (Show Source): You can put this solution on YOUR website!
.


Characteristics  "female"  and/or  "pathologist"  are certainly independent,  without any doubt,
it is why people invented these terms and use them in their everyday life.


But the events  "randomly select a female"  or  "randomly select a pathologist"  can be or can not be dependent or independent.



So,  learn how to formulate  Math problems correctly  (if you just started doing it).



Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

Given table
PathologyPediatricsPsychiatry
Male125753302027803
Female56043335112292

Compute the subtotals and the grand total
PathologyPediatricsPsychiatryTotal
Male12575330202780373398
Female5604333511229251247
Total181796637140095124645

Define the following events
F = doctor selected is female
A = doctor selected specializes in pathology

Based on the table, we can say
P(F) = (number of female doctors)/(number total)
P(F) = (51247)/(124645)
P(F) = 0.411144
Value is approximate to 6 decimal places

And also,
P(A) = (number of pathology doctors)/(number total)
P(A) = (18179)/(124645)
P(A) = 0.145846
Value is approximate to 6 decimal places

And,
P(F and A) = (number of female doctors in pathology)/(number total)
P(F and A) = (5604)/(124645)
P(F and A) = 0.044960
Value is approximate to 6 decimal places

Then note how
P(F)*P(A) = 0.411144*0.145846 = 0.059964
also approximate to 6 decimal places

But the result of P(F)*P(A) = 0.059964 does not match with P(F and A) = 0.044960

Since P(F and A) = P(F)*P(A) is a false equation, this means the events "selecting a female doctor" and "selecting a pathology doctor" are dependent.
The two groups are linked somehow.

P(F and A) = P(F)*P(A) must be a true equation for the groups to be independent.

Answer: Not independent (dependent)
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