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A certain multiple-choice math quiz contains five questions, each with

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A certain multiple-choice math quiz contains five questions, each with [#permalink]

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New post 23 Apr 2015, 03:43
3
11
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A
B
C
D
E

Difficulty:

  95% (hard)

Question Stats:

47% (01:35) correct 53% (01:52) wrong based on 122 sessions

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A certain multiple-choice math quiz contains five questions, each with four answer choices: A, B, C, and D. If the correct answer to each question is randomly distributed, which of the following represents the closest approximation of the probability that no two consecutive questions will have the same answer?

A. 25%
B. 32%
C. 40%
D. 48%
E. 75%


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Spoiler: OA

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A certain multiple-choice math quiz contains five questions, each with [#permalink]

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New post 23 Apr 2015, 13:33
1
Well lets see, something tells me it might be wrong but I'll give it a shot.
For pair 1-2
Total amount of choices for answers are 16: (AA, AB,AC,AD, BA,BB,BC,BD, CA,CB,CC,CD, DA,DB,DC,DD), total answers that give us bad result - 4( AA, BB, CC, DD), so odds of getting good outcome are 12/16 = 3/4
For pair 2-3, 3-4 and 4-5 its same 3/4.
All these events (questions have different answers) are dependant so the odds that 1-2 are different AND 2-3 are different ... AND 4-5 are different is pretty much (3/4)^4 = 0,316 ~ 32%

Answer ends up being "B"
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A certain multiple-choice math quiz contains five questions, each with [#permalink]

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New post 23 Apr 2015, 20:20
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Probability of no two consecutive question have same answer=4× 1/4 × 3/4 × 3/4 × 3/4 × 3/4 = 0.3164

Explanation : 1/4 = probability of selecting 'A'
3/4 = probability of not selecting 'A'. (b,d,c)

Answer 32% (approx )
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Re: A certain multiple-choice math quiz contains five questions, each with [#permalink]

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New post 24 Apr 2015, 02:24
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A certain multiple-choice math quiz contains five questions, each with four answer choices: A, B, C, and D. If the correct answer to each question is randomly distributed, which of the following represents the closest approximation of the probability that no two consecutive questions will have the same answer?

A. 25%
B. 32%
C. 40%
D. 48%
E. 75%

After every answer there is a chance of 3/4th that it will not be the same answer. Repeat this 4 times.

3/4 * 3/4 * 3/4 * 3/4 = 81/256 = approx 32%.

B.
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Re: A certain multiple-choice math quiz contains five questions, each with [#permalink]

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New post 27 Apr 2015, 02:33
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Bunuel wrote:
A certain multiple-choice math quiz contains five questions, each with four answer choices: A, B, C, and D. If the correct answer to each question is randomly distributed, which of the following represents the closest approximation of the probability that no two consecutive questions will have the same answer?

A. 25%
B. 32%
C. 40%
D. 48%
E. 75%


Kudos for a correct solution.


VERITAS PREP OFFICIAL SOLUTION:

This question feels like a doozy. Probability questions tend to give fits to even top-performing test-takers. You simply have to remember, however, what probability is:

number of desired outcomes divided by number of possible outcomes

Let's start out by finding the number of total outcomes. There are five questions on the quiz, so let's draw out five slots:

How many possible outcomes are there for the first slot? Well, given that there are four answer choices, A, B, C, and D, there are four possible outcomes.

4_ __ ____ ____ ____

The same will be true for all of the other questions, as each question has exactly four possible results: A, B, C, and D.

4_ _4 4_ 4_ _4

Since there are four possibilities for each of five slots, there are a total of 4 * 4 * 4 * 4 * 4 possibilities, or 4^5 possible outcomes. This number will go in the denominator of our probability fraction.

Now, let's focus on the number of desired outcomes. As it was before, the number of possibilities in the first slot will be four, as the correct answer to the first question could be A, B, C, or D.

4_ ____ ____ ____ ____

Here is where things get a bit tricky. The answer to the second question can be anything except the answer to the first. For example, if the answer to Question 1 is A, then Question 2 can only be B, C, or D.

If the answer to Question 1 is B, then Question 2 can only be A, C, or D. In any case, there are exactly three possibilities for the second slot.

4_ 3_ ____ ____ ____

Based on the answer to Question 2, there are three possibilities for Question 3 (because Question 3 cannot have the same answer as Question 2). Thus, there will also be three possibilities for the fourth and fifth slots, as their only restrictions are not having the same answer as the question immediately preceding it.

4_ 3_ 3_ 3_ 3_

Multiply these together, and you get 4 * 3 * 3 * 3 * 3, or 4 * 3^4 as the number of desired outcomes. So here is the probability fraction, and its subsequent reduction:

desired outcomes = 4 * 3^4

possible outcomes = 4^5

And since you're dividing them, you can factor out the common 4 to get a fraction of (3^4)/(4^4), which is 81/256.

Resist the temptation to do this math and figure out the exact percentage! You know that 80 / 240 = 33.333%, so 81/256 will be slightly less. The only answer that works is 32%, or answer choice B.
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Resources:
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Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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Re: A certain multiple-choice math quiz contains five questions, each with [#permalink]

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Re: A certain multiple-choice math quiz contains five questions, each with   [#permalink] 13 Jun 2018, 10:57
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