2024 AIME I Problems/Problem 4
Problem
Jen enters a lottery by picking distinct numbers from
numbers are randomly chosen from
She wins a prize if at least two of her numbers were
of the randomly chosen numbers, and wins the grand prize if all four of her numbers were the randomly chosen numbers. The probability of her winning the grand prize given that she won a prize is
where
are relatively prime. What is the value of
?
Solution 1
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
Solution 2
For getting all right, there is only
way.
For getting right, there is
multiplied by
=
.
For getting right, there is
multiplied by
=
.
=
Therefore, the answer is
~e___
Video Solution
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
See also
2024 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
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