Question

Each of $2010$ boxes in a line contains a single red marble, and for $1 \le k \le 2010,$ the box in the $k$ th position also contains $k$ white marbles. Isabella begins at the first box and successively draws a single marble at random from each box, in order. She stops when she first draws a red marble. Let $P(n)$ be the probability that Isabella stops after drawing exactly $n$ marbles. What is the smallest value of $n$ for which $P(n) \lt \dfrac{1}{2010}?$

$45$

$63$

$64$

$201$

$1005$

Answer: A

Concepts:telescoping basic probability inequality

Difficulty rating: 2240

Small Hint:

The first $n-1$ draws must be white, then the $n$ th draw red

Big Hint:

The white probabilities telescope: $\dfrac12\cdot\dfrac23\cdots\dfrac{n-1}{n}$

Solution:

Since there are $k + 1$ marbles in the $k$ th box, there is a $\dfrac{k}{k + 1}$ chance Isabella draws a white marble from it.

The probability of drawing a red marble is then $\dfrac{1}{k + 1}.$ To stop after drawing the $n$ th marble, the first $n - 1$ marbles must have been white.

This happens with a probability of $\dfrac{1}{2} \cdot \dfrac{2}{3} \cdot \ldots \cdot \dfrac{n - 1}{n} \cdot \dfrac{1}{n + 1}.$

Note that all the numerators cancel with the adjacent denominator, which means that this expression reduces to $\dfrac{1}{n(n + 1)}.$

We have to find the smallest $n$ such that $\dfrac{1}{n(n + 1)} \lt \dfrac{1}{2010}$ $n(n + 1) \gt 2010.$

Since $44\cdot45=1980\lt2010$ but $45\cdot46=2070\gt2010,$ the smallest such $n$ is $45.$

Thus, A is the correct answer.

2010 AMC 12A Problem 19

Answer: A

Difficulty rating: 2240

Small Hint:

Big Hint:

Solution:

Problem 19 in Other Years