Question

Hospital records show that 12% of all patients are admitted for surgical treatment (ST), 16% are admitted for obstetrical treatment (OT), and 2% receive both obstetrical and surgical treatment.

(a) Describe in terms of ST and OT the event that a newly admitted patient will receive at least one of these treatments and then compute its probability.

(b) Describe in terms of ST and OT the event that a newly admitted patient will receive surgical treatment but not obstetrical treatment and then compute its probability.

(c) Describe in terms of ST and OT the event that a newly admitted patient receives neither of these treatments and then compute its probability.

(d) If a newly admitted patient receives obstetrical treatment, what is the probability that the patient also receives surgical treatment?

(e) Are OT and ST independent events? Explain.

Answer 1

Let

• ST=event that a new patient receives surgical treatment and is the event that a new patient does not receive surgical treatment
• OT=event that a new patient receives obstetrical treatment and is the event that a new patient does not receive obstetrical treatment

We know the following probabilities

a)  the event that a newly admitted patient will receive at least one of these treatments is the event that a newly admitted patient will receive surgical treatment (ST) or obstetrical treatment (OT).

ans:  

The probability that a newly admitted patient will receive at least one of these treatments is

ans: The probability that a newly admitted patient will receive at least one of these treatments is 0.26

b)  the event that a newly admitted patient will receive surgical treatment but not obstetrical treatment is

ans:

The probability that a newly admitted patient will receive surgical treatment but not obstetrical treatment is

ans: The probability that a newly admitted patient will receive surgical treatment but not obstetrical treatment is 0.10

c) the event that a newly admitted patient receives neither of these treatments is

ans:

The probability that a newly admitted patient receives neither of these treatments is

ans: The probability that a newly admitted patient receives neither of these treatments is 0.74

d) If a newly admitted patient receives obstetrical treatment, the probability that the patient also receives surgical treatment is same as the conditional probability that a newly admitted patient receives surgical treatment given that the patient receives obstetrical treatment  and it is

ans: If a newly admitted patient receives obstetrical treatment, the probability that the patient also receives surgical treatment is 0.125

e) We know that if events OT and St are independent then

We will use the result of conditional probability from part d) and can say that

Ans: OT and ST are not independent.

We could also check

and hence OT and ST are not independent.