2024 AIME I Problems/Problem 4
This is a conditional probability problem. Bayes' Theorem states that
- in other words, the probability of
given
is equal to the probability of
given
times the probability of
divided by the probability of
. In our case,
represents the probability of winning the grand prize, and
represents the probability of winning a prize. Clearly,
, since by winning the grand prize you automatically win a prize. Thus, we want to find
.
Let us calculate the probability of winning a prize. We do this through casework: how many of Jen's drawn numbers match the lottery's drawn numbers?
To win a prize, Jen must draw at least numbers identical to the lottery. Thus, our cases are drawing
,
, or
numbers identical.
Let us first calculate the number of ways to draw exactly identical numbers to the lottery. Let Jen choose the numbers
,
,
, and
; we have
ways to choose which
of these
numbers are identical to the lottery. We have now determined
of the
numbers drawn in the lottery; since the other
numbers Jen chose can not be chosen by the lottery, the lottery now has
numbers to choose the last
numbers from. Thus, this case is
, so this case yields
possibilities.
Next, let us calculate the number of ways to draw exactly identical numbers to the lottery. Again, let Jen choose
,
,
, and
. This time, we have
ways to choose the identical numbers and again
numbers left for the lottery to choose from; however, since
of the lottery's numbers have already been determined, the lottery only needs to choose
more number, so this is
. This case yields
.
Finally, let us calculate the number of ways to all numbers matching. There is actually just one way for this to happen.
In total, we have ways to win a prize. The lottery has
possible combinations to draw, so the probability of winning a prize is
. There is actually no need to simplify it or even evaluate
or actually even know that it has to be
; it suffices to call it
or some other variable, as it will cancel out later. However, let us just go through with this. The probability of winning a prize is
. Note that the probability of winning a grand prize is just matching all
numbers, which we already calculated to have
possibility and thus have probability
. Thus, our answer is
. Therefore, our answer is
.
~Technodoggo