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Doctor Specialties Below are listed the numbers of doctors in various specialties by gender.
Pathology | Pediatrics | Psychiatry | |
Male | 12,575 | 33,020 | 27,803 |
Female | 5,604 | 33,351 | 12,292 |
Choose one doctor at random.
a. Find P (male|pediatrician).
b. Find P (pathologist |female).
c. Are the characteristics “female” and “pathologist” independent? Explain.
Source: World Almanac.
a.
![Check Mark](/static/check-mark.png)
To obtain: The probability that the doctor was male given that the doctor was pediatrician.
Answer to Problem 32E
The probability that the doctor was male given that the doctor was pediatrician is 0.498.
Explanation of Solution
Given info:
The data shows that the the numbers of doctors in various specialties by gender.
Calculation:
The total number of genders is shown in the Table (1).
Pathology | Pediatrics | Psychiatry | Total | |
Male | 12,575 | 33,020 | 27,803 | 73,398 |
Female | 5,604 | 33,351 | 12,292 | 51,247 |
Total | 18,179 | 66,371 | 40,095 | 124,645 |
Table (1)
Let event A denote that the doctor was pediatrician and event B denote that the doctor was male.
The formula for probability of event A is,
Substitute 66,371 for ‘Number of outcomes in A’ and 124,645 for ‘Total number of outcomes in the sample space’,
The formula for probability of event A and B is,
Substitute 33,020 for ‘Number of outcomes in A and B’ and 124,645 for ‘Total number of outcomes in the sample space’,
Conditional rule:
The formula for probability of A given B is,
Substitute 0.265 for ‘
The required probability is,
Thus, the probability that the doctor was male given that the doctor was pediatrician is 0.498.
b.
![Check Mark](/static/check-mark.png)
To obtain: The probability that the doctor was pathologist given that the doctor was female.
Answer to Problem 32E
The probability that the doctor was pathologist given that the doctor was female is 0.109.
Explanation of Solution
Calculation:
Let event C denote that the doctor was female and event D denote that the doctor was pathologist.
The probability of event C is,
Substitute 51,247 for ‘Number of outcomes in C’ and 124,645 for ‘Total number of outcomes in the sample space’,
The formula for probability of event C and D is,
Substitute 5,604 for ‘Number of outcomes in C and D’ and 124,645 for ‘Total number of outcomes in the sample space’,
Applying the conditional probability, then the required probability is,
Thus, the probability that the doctor was pathologist given that the doctor was female is 0.109.
c.
![Check Mark](/static/check-mark.png)
To explain: Whether the characteristics “female” and “pathologist” independent or not.
Answer to Problem 32E
The characteristics “female” and “pathologist” are not independent because
Explanation of Solution
Calculation:
Here, the event C denotes that the doctor was female and event D denotes that the doctor was pathologist.
From part (b), it is observed that the probability that the doctor was female is 0.411 and
The probability of event D is,
Substitute 18,179 for ‘Number of outcomes in D’ and 124,645 for ‘Total number of outcomes in the sample space’,
Independent events:
If
Substitute 0.411for
Here,
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Chapter 4 Solutions
Elementary Statistics: A Step By Step Approach
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