Intuitive understanding of the "Multiplication Rule"?

Question

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I apologize in advance that this question has a long set-up. In the set up I am presenting how I currently understand the material, and the actual question is if my understanding is correct and applicable to more types of questions.

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I am referring to the rule $$P(A \text{ and } B)=P(A) \cdot P(B|A)$$ Or just $$P(A \text{ and } B)=P(A) \cdot P(B)$$

If the events are independent.

I will be using absolute value bars as notation for the size of a set (i.e. $|\{z_1, z_2, ...z_n\}|=n$)

I feel like I understand this rule for the most part when we have $2$ independent events. For example, rolling an even number on a standard dice and flipping "tails" with a coin. The way I understand this is as follows:

$$ \dfrac{ \text{Desired outcomes}}{\text{Total Outcomes}} = \dfrac{|\{2T, 4T, 6T\}|}{|\{1H, 1T, 2H, 2T,...,6H, 6T\}|} = \dfrac{|\{2, 4, 6 \}|}{|\{1, 2, 3, 4, 5, 6\}|} \cdot \dfrac{ |\{T\}|}{|\{H, T\}|}$$

When it comes to two dependent events, it mostly makes sense but I don't think I've completely internalized it. For example, if the question asks:

"Find the probability of drawing a king, and then a queen from a deck of cards without replacement."

I have a little bit of trouble making sense of this one in the context of one deck, so I just pretend that we have two decks, one with $52$ cards and the other with $51$ cards (with the king of diamonds missing), and I reduce the problem to drawing a king and a queen in sequence from the two decks. I think I can make sense of it the same way I made sense of the dice and the coin:

$$ \dfrac{ \text{Desired outcomes}}{\text{Total Outcomes}} = \dfrac{|\{K_{clubs}Q_{clubs}, K_{clubs}Q_{spades},...,K_{diamonds}Q_{diamonds}\}|}{|\{Ace_{clubs}Ace_{clubs}, Ace_{clubs}Ace_{spades},..., K_{diamonds}K_{hearts} \}|} = \dfrac{|\{K_{clubs}, K_{spades}, K_{hearts}, K_{diamond} \}|}{|\{Ace_{clubs}, Ace_{spades},...K_{diamonds}\}|} \cdot \dfrac{ |\{ Q_{clubs}, ... Q_{diamond} \}|}{| \{Ace_{clubs}, Ace_{spades},...K_{hearts} \}|}$$

However, what really gives me trouble is when we have just one event. For example

"15,000 U.S. medical school seniors applied to residency programs in 2009. Of those, 93% were matched with residency positions. 82% of those seniors matched with residency positions were matched with one of their top $3$ choices. Find the probability that a randomly selected student was matched with a residency position and that it was one of their top $3$ choices."

This problem is easy to solve from an "algebraic":

$$\dfrac{.82 \cdot(\text{n. of students who got a residency position})}{\text{total number of students}}=\dfrac{.82(.93 \cdot 15,000)}{15,000} $$

However, when I tried to tackle this one like I did the card problem, everything collapsed. Without loss of generality, we let students $s_1$ to $s_{13950}$ be the students who got residency positions, and let students $s_1$ to $s_{11439}$ be the students who got matched with one of their top $3$ choices. If I try to multiply like I did with the cards problem, the results just don't make sense to me (even though I know it gives me the right answer):

$$\dfrac{|\{s_1,...,s_{13950}\}|}{|\{s_1,...,s_{15000}\}|} \cdot \dfrac{|\{s_1,...,s_{11439}\}|}{|\{ s_1,...s_{13950} \}|} = \dfrac{|\{s_1s_1, s_1s_2,...,s_{13950}s_{11439} \}|}{|\{s_1s_1,...,s_{15000}s_{13950}\}|}$$

My questions are: Is my understanding fine for independent events? Can it be applied to dependent events (like with the card problem)? Can it be applied to problems containing only $1$ choice (like the last problem)?

Answer 1

Your frequentist understanding will produce correct results for independent discrete uniform probability distributions, but it simply can't be transferred to dependent events, which seems to be the problem you are having here. Taking the cartesian product can simply not be transferred to the dependent case, since the elements of the cartesian product no longer all have the same probability. For example the event $s_{15000}s_{13950}$ has probability exactly 0, since the student you picked can't be number 15000 and number 13950 at the same time.

Attempting to solve mathematical problems by intuition will stop working as soon as the problems become less intuitive than you are used to. I suggest that instead of trying to keep interpreting probabilities as relative frequencies, you learn to work with them directly, using the axioms, theorems and rules of probability.

In this case you should recognize the numbers you are given as the probabilities that they are, so you can insert them into the multiplication rule as it is stated. If $s$ is a randomly selected student, and $A$ denotes the event "$s$ was matched with a residency position", and $B$ denotes the event "$s$ was matched with one of their top 3 choices", then which numbers were you given?

The 93% is $P(A)$, and the 82% is $P(B|A)$, and the 15000 is irrelevant. You can now apply the multiplication rule to get $P(A\cap B)$, which is exactly the probability the problem demands you find out, and you didn't have to imagine 15000 students and calculate numbers such as 13950.

Answer 2

The way you seem to be intuitively understanding the probability of two events both occurring sometimes makes sense, but sometimes seems to make things more complicated. I'm not so sure that it simply depends on whether we are dealing with independent or dependent events.

In your two card example, you can look at the number of possible king-queen outcomes and the total number of two card outcomes, and divide to get the desired probability. In this case the two events (1st card is King, 2nd card is Queen) are dependent, but your understanding is correct. (One thing that I would change though is that I wouldn't list something like $A_{clubs}A_{clubs}$ in the possible outcome pool as this is not actually a possible outcome.)

In the medical school example, your method would still work but it doesn't seem to be the simplest way of looking at it. The understanding behind your formula would be that in the 1st event we are selecting a student from a pool of all 15,000 students, and in the 2nd event we are selecting a student from a pool of the 13,950 students with residency. The desired combinations are where the 1st student got a residency and the 2nd got a choice match and we compare the desired combinations with the total combinations to get the desired probability. While this is technically accurate, the more intuitive understanding would be that we are just making one selection from the 15,000 students with the desired outcome being a student who got a residency with a top choice and the desired probability being (# of desired outcomes/# of total outcomes).

Your way of understanding joint probabilities is sometimes not so intuitive even with some independent events. For example, if I would say "40% of students in the class were male and 90% of the males in the class got their residency of choice". The two events may very well be independent. Regardless, it would be more intuitive to view the probability as (males who got their residency/total students) rather than looking at the # of possible combinations of (student, male student) compared to the # of possible combinations of (male student, male student with residency of choice).

In short, the intuitive understanding may vary based on the scenario, but the formula remains the same.