Question

question
let ( c ) be the event that a randomly chosen cancer patient has received chemotherapy. let ( e ) be the event that a randomly chosen cancer patient has received elective surgery. identify the answer which expresses the following with correct notation: of all the cancer patients that have received chemotherapy, the probability that a randomly chosen cancer patient has had elective surgery.

select the correct answer below:
( p(c|e) )
( p(e|c) )
( p(e) ) and ( p(c) )
( p(c \text{ and } e) )

Explanation:

To determine the correct notation, we recall the definition of conditional probability. The notation \( P(A|B) \) represents the probability of event \( A \) occurring given that event \( B \) has already occurred. In this problem, we want the probability of a patient having had elective surgery (\( E \)) given that they have received chemotherapy (\( C \)). So we need the probability of \( E \) given \( C \), which is denoted as \( P(E|C) \).

• \( P(C|E) \) would be the probability of receiving chemotherapy given elective surgery, which is not what we want.
• \( P(E) \) AND \( P(C) \) is not a valid conditional probability notation for this context.
• \( P(C \text{ AND } E) \) is the probability of both events occurring, not a conditional probability.

Answer:

B. \( P(E|C) \)