Difference between revisions of "2010 AIME I Problems/Problem 4"
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The generating functions for these coins are <math>(1+x)</math>,<math>(1+x)</math>,and <math>(4+3x)</math> in order. | The generating functions for these coins are <math>(1+x)</math>,<math>(1+x)</math>,and <math>(4+3x)</math> in order. | ||
− | The product is <math> | + | The product is <math>4+11x+10x^2+3x^3</math>. (<math>ax^n</math> means there are <math>a</math> ways to get <math>n</math> heads, eg there are <math>11</math> ways to get <math>2</math> heads, and therefore <math>1</math> tail, here.) |
The sum of the coefficients squared is <math>784</math> and the sum of the squares of each coefficient is <math>246</math>. | The sum of the coefficients squared is <math>784</math> and the sum of the squares of each coefficient is <math>246</math>. |
Revision as of 22:12, 17 May 2012
Problem
Jackie and Phil have two fair coins and a third coin that comes up heads with probability . Jackie flips the three coins, and then Phil flips the three coins. Let
be the probability that Jackie gets the same number of heads as Phil, where
and
are relatively prime positive integers. Find
.
Solution
Solution 1
This can be solved quickly and easily with generating functions.
Let represent flipping
tails.
The generating functions for these coins are ,
,and
in order.
The product is . (
means there are
ways to get
heads, eg there are
ways to get
heads, and therefore
tail, here.)
The sum of the coefficients squared is and the sum of the squares of each coefficient is
.
The probability is then
.
Solution 2
We perform casework based upon the number of heads that are flipped.
- Case 1: No heads.
- The only possibility is TTT (the third coin being the unfair coin). The probability for this to happen to Jackie is
Thus the probability for this to happen to both players is
- Case 2: One head.
- We can have either HTT, THT, or TTH. The first two happen to Jackie with the same
chance, but the third happens
of the time, since the unfair coin is heads instead of tails. With 3 possibilities for Jackie and 3 for Phil, there are a total of 9 ways for them both to have 1 head.
- Multiplying and adding up all 9 ways, we have a
- overall chance for this case.
- Case 3: Two heads.
- With HHT
, HTH
, and THH
possible, we proceed as in Case 2, obtaining
![\[\frac {1(3 \cdot 3) + 4(3 \cdot 4) + 4(4 \cdot 4)}{28^{2}} = \frac {121}{784}.\]](http://latex.artofproblemsolving.com/7/4/9/7499bc0145c48d7afd1373e83db7cf746fc18412.png)
- Case 4: Three heads.
- Similar to Case 1, we can only have HHH, which has
chance. Then in this case we get
Finally, we take the sum: , so our answer is
.
See Also
2010 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 | ||
All AIME Problems and Solutions |