Difference between revisions of "2010 AMC 12A Problems/Problem 19"

Revision as of 22:25, 16 February 2014

Problem

Each of 2010 boxes in a line contains a single red marble, and for $1 \le k \le 2010$ , the box in the $k\text{th}$ 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 $P(n) < \frac{1}{2010}$ ?

$\textbf{(A)}\ 45 \qquad \textbf{(B)}\ 63 \qquad \textbf{(C)}\ 64 \qquad \textbf{(D)}\ 201 \qquad \textbf{(E)}\ 1005$

Solution

Solution 1

The probability of drawing a white marble from box $k$ is $\frac{k}{k+1}$ . The probability of drawing a red marble from box $n$ is $\frac{1}{n+1}$ .

The probability of drawing a red marble at box $n$ is therefore

$\frac{1}{n+1} \left( \prod_{k=1}^{n-1}\frac{k}{k+1} \right) < \frac{1}{2010}$

$\frac{1}{n+1} \left( \frac{1}{n} \right) < \frac{1}{2010}$

$(n+1)n > 2010$

It is then easy to see that the lowest integer value of $n$ that satisfies the inequality is $\boxed{45\ \textbf{(A)}}$ .

Solution 2

The probability of drawing a white marble from the first box is $\frac{1}{2}$ . The probability of drawing a white marble from the second box is $\frac{2}{3}$ . IT follows that the probability of drawing a white marble from box $k$ is $\frac{k}{k + 1}$ , and the probability of drawing a white marble is $\frac{1}{k+1}$ .

From this, we find that $P(n) = (\frac{1}{2} \cdot \frac{2}{3} \cdot \frac{3}{4} \cdot \dots \cdot \frac {n - 1}{n}) \cdot \frac{1}{n +1}$

Clearly, $P(n) = \frac{(n - 1)!}{(n + 1)!} = \frac{1}{n (n + 1)}$

So $\frac{1}{n(n + 1)} < \frac{1}{2010}$ $n(n+1) > 2010$

The minimum integer value of $n$ is $\boxed{\textbf{(A)}45}$ .

@@ Line 5: / Line 5: @@
 == Solution ==
+===Solution 1===
 The probability of drawing a white marble from box <math>k</math> is <math>\frac{k}{k+1}</math>. The probability of drawing a red marble from box <math>n</math> is <math>\frac{1}{n+1}</math>.
@@ Line 18: / Line 19: @@
 It is then easy to see that the lowest integer value of <math>n</math> that satisfies the inequality is <math>\boxed{45\ \textbf{(A)}}</math>.
+===Solution 2===
+The probability of drawing a white marble from the first box is <math>\frac{1}{2}</math>. The probability of drawing a white marble from the second box is <math>\frac{2}{3}</math>. IT follows that the probability of drawing a white marble from box <math>k</math> is <math>\frac{k}{k + 1}</math>, and the probability of drawing a white marble is <math>\frac{1}{k+1}</math>.
+From this, we find that <cmath>P(n) = (\frac{1}{2} \cdot \frac{2}{3} \cdot \frac{3}{4} \cdot \dots \cdot \frac {n - 1}{n}) \cdot \frac{1}{n +1}</cmath>
+Clearly, <cmath>P(n) = \frac{(n - 1)!}{(n + 1)!} = \frac{1}{n (n + 1)}</cmath>
+So <cmath>\frac{1}{n(n + 1)} < \frac{1}{2010}</cmath> <cmath>n(n+1) > 2010</cmath>
+The minimum integer value of <math>n</math> is <math>\boxed{\textbf{(A)}45}</math>.
 == See also ==

2010 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 18	Followed by Problem 20
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

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