# Probability of Guesser Scoring 100% on Midvale School Tests

• Philip Wong
In summary, the Midvale School for the Gifted has two types of students: Guessers and Swots. Guessers have a 50% chance of getting each question correct, while Swots will always get every question correct. The Chief Examiner, Dr Ramble Fluster, is concerned about test results and must decide how many questions to include in each test to ensure that she can give the correct grade (G or S) to each student. The probability of a Guesser scoring 100% on a test with n questions is nCn * 0.5^n * 0.5^0. To guarantee that there is less than a 0.5% chance of a Guesser getting full marks, the lowest
Philip Wong

## Homework Statement

Midvale School for the Gifted has two types of students: Guessers and Swots. All
Midvale tests consist of sets of questions with yes/no answers. Guessers will simply
answer yes or no to each question as the mood takes them, so they have probability
0.5 of getting each question correct. Swots, on the other hand, will certainly get
every question correct with probability 1. Dr Ramble Fluster is the Chief Examiner
at Midvale. She worries about her test results. Clearly, all Swots will get 100% on
every test. However, it is possible that a Guesser might get 100% just by chance.
Ramble's job is to decide how many questions to set in each test to be fairly certain
that she will be able to give the correct grade (G or S) to each student.

(b) What is the probability that a Guesser scores 100% on a test with n questions?

## Homework Equations

Binomial function

## The Attempt at a Solution

X~Bin(n,0.5)

P(X=x)= nCx * p^x * (1-p)^(n-x)

P(X=100) = nC100 * 0.5^100 * 0.5^(n-100)

here is where I got stuck. Since the question is asking us to solve the probability equation (P(X=x)), but since n is an unknown I don't know how to go further into solving this equation, because (I presume) there will be 2 unknown parameters: P and n.

can someone point me where I get wrong, or give me an idea on how to solve this?
thanks

nC100?? Think about what that means. You aren't choosing 100 percentage points from n percentage points. Scoring 100% means you got all n questions right. Shouldn't the probability involve nCn?

Dick said:
nC100?? Think about what that means. You aren't choosing 100 percentage points from n percentage points. Scoring 100% means you got all n questions right. Shouldn't the probability involve nCn?

ar! ok, I've just reattempt this question from what I understand in your suggestion. Which doesn't seems to be correct though, need more help thanks.

reattempts are as follows:

fx(x)= p(X=x) = nCx * p^x * (1-p)^n-x

let x = n

p(X=n) = nCn * p ^ n * (1-p)^n-n
= 1 * 0.5^n * 0.5^0
= 0.5^n

Philip Wong said:
ar! ok, I've just reattempt this question from what I understand in your suggestion. Which doesn't seems to be correct though, need more help thanks.

reattempts are as follows:

fx(x)= p(X=x) = nCx * p^x * (1-p)^n-x

let x = n

p(X=n) = nCn * p ^ n * (1-p)^n-n
= 1 * 0.5^n * 0.5^0
= 0.5^n

That seems ok to me. E.g. if there are two questions P=(1/4)=(0.5)^2. You have to get the first question right AND the second question right - both with probability 1/2.

Dick said:
That seems ok to me. E.g. if there are two questions P=(1/4)=(0.5)^2. You have to get the first question right AND the second question right - both with probability 1/2.

ah! looks light you are right. because I just read the next part of the question "Ramble wishes to select the number of questions, n, so that there is less than 0.5% chance that a Guesser gets full marks. What is the lowest value of n that will guarantee this?"

so I will just substitute in the numbers and found out what n is.

thanks a lot for your help!

You're welcome. You could solve it systematically using a log. But there's nothing wrong with substituting either.

## 1. What is the probability of a guesser scoring 100% on Midvale School Tests?

The probability of a guesser scoring 100% on Midvale School Tests is extremely low, close to 0%. This is because guessing would only result in a correct answer about 25% of the time, assuming there are four answer choices for each question.

## 2. How does the difficulty level of the test affect the probability of scoring 100%?

The difficulty level of the test has a direct impact on the probability of scoring 100%. The more difficult the test, the lower the probability since the guesser would have a harder time getting the correct answer by chance.

## 3. Can someone improve their chances of scoring 100% by guessing on multiple choice tests?

No, guessing on multiple choice tests does not improve the chances of scoring 100%. In fact, it may decrease the chances since incorrect guesses would result in a lower overall score.

## 4. Are there any factors that could increase the probability of scoring 100% on Midvale School Tests?

Yes, there are factors that could increase the probability of scoring 100% on Midvale School Tests. These include being familiar with the material, studying and preparing for the test, and having a higher level of intelligence and critical thinking skills.

## 5. Is it possible for a guesser to score 100% on Midvale School Tests?

Technically, it is possible for a guesser to score 100% on Midvale School Tests, but it is highly improbable. It would require an incredible amount of luck and would be nearly impossible to achieve consistently.

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