Question

course: probability and statistics
in a recent year, there were 20,187 u.s. allopathic medical school seniors who applied to residency programs and submitted their residency program choices. of these seniors, 18,200 were matched with residency positions, with about 89.2% getting one of their top three choices. medical students rank the residency programs in their order of preference, and program directors in the united states rank the students. the term \match\ refers to the process whereby a students preference list and a program directors preference list overlap, resulting in the placement of the student in a residency position. (source: national resident matching program
1 - find the probability that a randomly selected senior was matched with a residency position and it was one of the seniors top three choices
2 - find the probability that a randomly selected senior that was matched to a residency position did not get matched with one of the seniors top three choices
3 - give two example of independent events and two examples of dependent examples

Explanation:

Step1: Probability of being matched and in top three

Let \(A\) be the event of being matched and \(B\) be the event of being in top - three.
We know \(n = 20187\) (total number of seniors), \(n(A)=18200\) (number of matched seniors), and \(P(B|A)=0.892\) (probability of being in top - three given matched).
By the formula \(P(A\cap B)=P(B|A)P(A)\), and \(P(A)=\frac{n(A)}{n}=\frac{18200}{20187}\approx0.902\).
So \(P(A\cap B)=0.892\times\frac{18200}{20187}\approx0.892\times0.902 = 0.804\).

Step2: Probability of not in top three given matched

Let \(C\) be the event of not being in top - three given matched.
Since \(P(B|A) + P(C|A)=1\) (complementary events).
Given \(P(B|A)=0.892\), then \(P(C|A)=1 - 0.892=0.108\).

Step3: Independent and dependent events

• Independent events:
• Tossing a coin twice. The outcome of the first toss (\(E_1\)) does not affect the outcome of the second toss (\(E_2\)). \(P(E_1\cap E_2)=P(E_1)P(E_2)\).
• Rolling a die and then flipping a coin. The result of rolling the die (\(E_3\)) has no impact on the result of flipping the coin (\(E_4\)). \(P(E_3\cap E_4)=P(E_3)P(E_4)\).
• Dependent events:
• Drawing two cards from a deck without replacement. Let \(E_5\) be drawing a king first and \(E_6\) be drawing a queen second. \(P(E_6|E_5)

eq P(E_6)\) (because after drawing a king, the number of cards in the deck changes).

• Selecting students from a class for a project. Let \(E_7\) be selecting a boy first and \(E_8\) be selecting a girl second. The probability of \(E_8\) depends on whether \(E_7\) has occurred (the composition of the remaining students changes).

Answer:

1. The probability that a randomly selected senior was matched with a residency position and it was one of the senior's top three choices is approximately \(0.804\).
2. The probability that a randomly selected senior that was matched to a residency position did not get matched with one of the senior's top three choices is \(0.108\).

3.

• Independent events: Tossing a coin twice; Rolling a die and then flipping a coin.
• Dependent events: Drawing two cards from a deck without replacement; Selecting students from a class for a project.