Difference between revisions of "2006 iTest Problems/Problem 12"

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==Solution==
 
==Solution==
The highest probability occurs when the first <math>8</math> problems are correct. The probability is thus <cmath>\frac{1}{1}\cdot\frac{1}{2}\cdot\frac{1}{3}\dots\frac{1}{8}\cdot\frac{8}{9}\cdot\frac{9}{10}\dots\frac{18}{19}\cdot\frac{19}{20}</cmath> This, when simplified, gives <cmath>\frac{1}{7!\cdot20}</cmath> Which is in none of the answer choices, so the answer is <math>\boxed{(\text{L})}</math>
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The highest probability occurs when the first <math>8</math> problems are correct. The probability is thus <cmath>\frac{1}{1}\cdot\frac{1}{2}\cdot\frac{1}{3}\dots\frac{1}{8}\cdot\frac{8}{9}\cdot\frac{9}{10}\dots\frac{18}{19}\cdot\frac{19}{20}</cmath> This, when simplified, gives <cmath>\frac{1}{7!\cdot20}</cmath> Which is in none of the answer choices, so the answer is <math>\boxed{\text{(L) }\text{none of the above}}</math>
  
 
==See Also==
 
==See Also==
  
 
{{iTest box|year=2006|num-b=11|num-a=13|ver=[[2006 iTest Problems/Problem U1|U1]] '''•''' [[2006 iTest Problems/Problem U2|U2]] '''•''' [[2006 iTest Problems/Problem U3|U3]] '''•''' [[2006 iTest Problems/Problem U4|U4]] '''•''' [[2006 iTest Problems/Problem U5|U5]] '''•''' [[2006 iTest Problems/Problem U6|U6]] '''•''' [[2006 iTest Problems/Problem U7|U7]] '''•''' [[2006 iTest Problems/Problem U8|U8]] '''•''' [[2006 iTest Problems/Problem U9|U9]] '''•''' [[2006 iTest Problems/Problem U10|U10]]}}
 
{{iTest box|year=2006|num-b=11|num-a=13|ver=[[2006 iTest Problems/Problem U1|U1]] '''•''' [[2006 iTest Problems/Problem U2|U2]] '''•''' [[2006 iTest Problems/Problem U3|U3]] '''•''' [[2006 iTest Problems/Problem U4|U4]] '''•''' [[2006 iTest Problems/Problem U5|U5]] '''•''' [[2006 iTest Problems/Problem U6|U6]] '''•''' [[2006 iTest Problems/Problem U7|U7]] '''•''' [[2006 iTest Problems/Problem U8|U8]] '''•''' [[2006 iTest Problems/Problem U9|U9]] '''•''' [[2006 iTest Problems/Problem U10|U10]]}}

Latest revision as of 22:43, 3 November 2023

Problem

What is the highest possible probability of getting $12$ of these $20$ multiple choice questions correct, given that you don't know how to work any of them and are forced to blindly guess on each one?

$\text{(A) }\frac{1}{6!}\qquad \text{(B) }\frac{1}{7!}\qquad \text{(C) }\frac{1}{8!}\qquad \text{(D) }\frac{1}{9!}\qquad \text{(E) }\frac{1}{10!}\qquad \text{(F) }\frac{1}{11!}\qquad\\ \\ \text{(G) }\frac{1}{12!}\qquad \text{(H) }\frac{2}{8!}\qquad \text{(I) }\frac{2}{10!}\qquad \text{(J) }\frac{2}{12!}\qquad \text{(K) }\frac{1}{20!}\qquad \text{(L) }\text{none of the above}\qquad$

(Clarification: the $n\text{th}$ question has $n$ answer choices, where $n$ goes from $1$ to $20$)

Solution

The highest probability occurs when the first $8$ problems are correct. The probability is thus \[\frac{1}{1}\cdot\frac{1}{2}\cdot\frac{1}{3}\dots\frac{1}{8}\cdot\frac{8}{9}\cdot\frac{9}{10}\dots\frac{18}{19}\cdot\frac{19}{20}\] This, when simplified, gives \[\frac{1}{7!\cdot20}\] Which is in none of the answer choices, so the answer is $\boxed{\text{(L) }\text{none of the above}}$

See Also

2006 iTest (Problems, Answer Key)
Preceded by:
Problem 11
Followed by:
Problem 13
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 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 U1 U2 U3 U4 U5 U6 U7 U8 U9 U10