Difference between revisions of "2009 AIME II Problems/Problem 8"

m (Solution 2)
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With probability <math>\frac 1{36}</math>, both throw a six and we win.
 
With probability <math>\frac 1{36}</math>, both throw a six and we win.
  
With probability <math>\frac{10}{36}</math> exactly one of them throws a six. In this case, we win iff the remaining player throws a six in their next throw, which happens with probability <math>\frac 16</math>.
+
With probability <math>\frac{10}{36}</math> exactly one of them throws a six. In this case, we win if the remaining player throws a six in their next throw, which happens with probability <math>\frac 16</math>.
  
 
Finally, with probability <math>\frac{25}{36}</math> none of them throws a six. Now comes the crucial observation: At this moment, we are in exactly the same situation as in the beginning. Hence in this case we will win with probability <math>p</math>.
 
Finally, with probability <math>\frac{25}{36}</math> none of them throws a six. Now comes the crucial observation: At this moment, we are in exactly the same situation as in the beginning. Hence in this case we will win with probability <math>p</math>.
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Solving for <math>p</math>, we get <math>p=\frac 8{33}</math>, hence the answer is <math>8+33 = \boxed{041}</math>.
 
Solving for <math>p</math>, we get <math>p=\frac 8{33}</math>, hence the answer is <math>8+33 = \boxed{041}</math>.
 
  
 
== See Also ==
 
== See Also ==

Revision as of 19:09, 31 May 2020

Problem

Dave rolls a fair six-sided die until a six appears for the first time. Independently, Linda rolls a fair six-sided die until a six appears for the first time. Let $m$ and $n$ be relatively prime positive integers such that $\dfrac mn$ is the probability that the number of times Dave rolls his die is equal to or within one of the number of times Linda rolls her die. Find $m+n$.

Solution

Solution 1

There are many almost equivalent approaches that lead to summing a geometric series. For example, we can compute the probability of the opposite event. Let $p$ be the probability that Dave will make at least two more throws than Linda. Obviously, $p$ is then also the probability that Linda will make at least two more throws than Dave, and our answer will therefore be $1-2p$.

How to compute $p$?

Suppose that Linda made exactly $t$ throws. The probability that this happens is $(5/6)^{t-1}\cdot (1/6)$, as she must make $t-1$ unsuccessful throws followed by a successful one. In this case, we need Dave to make at least $t+2$ throws. This happens iff his first $t+1$ throws are unsuccessful, hence the probability is $(5/6)^{t+1}$.

Thus for a fixed $t$ the probability that Linda makes $t$ throws and Dave at least $t+2$ throws is $(5/6)^{2t} \cdot (1/6)$.

Then, as the events for different $t$ are disjoint, $p$ is simply the sum of these probabilities over all $t$. Hence:

\begin{align*} p & = \sum_{t=1}^\infty \left(\frac 56\right)^{2t} \cdot \frac 16 \\ & = \frac 16 \cdot \left(\frac 56\right)^2 \cdot \sum_{x=0}^\infty \left(\frac{25}{36}\right)^x \\ & = \frac 16 \cdot \frac{25}{36} \cdot \frac 1{1 - \dfrac{25}{36}} \\ & = \frac 16 \cdot \frac{25}{36} \cdot \frac{36}{11} \\ & = \frac {25}{66} \end{align*}

Hence the probability we were supposed to compute is $1 - 2p = 1 - 2\cdot \frac{25}{66} = 1 - \frac{25}{33} = \frac 8{33}$, and the answer is $8+33 = \boxed{041}$.

Solution 2

Let $p$ be the probability that the number of times Dave rolls his die is equal to or within one of the number of times Linda rolls her die. (We will call this event "a win", and the opposite event will be "a loss".)

Let both players roll their first die.

With probability $\frac 1{36}$, both throw a six and we win.

With probability $\frac{10}{36}$ exactly one of them throws a six. In this case, we win if the remaining player throws a six in their next throw, which happens with probability $\frac 16$.

Finally, with probability $\frac{25}{36}$ none of them throws a six. Now comes the crucial observation: At this moment, we are in exactly the same situation as in the beginning. Hence in this case we will win with probability $p$.

We just derived the following linear equation: \[p = \frac 1{36} + \frac{10}{36} \cdot \frac 16 + \frac{25}{36} \cdot p\]

Solving for $p$, we get $p=\frac 8{33}$, hence the answer is $8+33 = \boxed{041}$.

See Also

2009 AIME II (ProblemsAnswer KeyResources)
Preceded by
Problem 7
Followed by
Problem 9
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions

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