True/False Test

Probability Level 3

On a True/False test tomorrow, you will be given 20 True/False questions.

Your instructor tells you that you can take the test as many times as you like but the questions (and their corresponding answers) will be changed each time you decide to re-take the test.

Further you can judge your score instantly after submitting your test and has the choice to re-take the test if you are not satisfied with your score.

Seeing this opportunity, you decide to give the test unprepared. You are going to give the test and re-take it as many times as you don't score greater than or equal to 15.

Each question carries a score of 1. There's no negative marking. The probability that the answer to a particular question is True is 1/2 and is independent of answer to any other question. Also assume you know nothing and tick True with probability 1/2.

What's the expected number of times you have to take the test inorder to have a score equal to or greater than 15:

Around 70 Around 80 Around 50 Around 60

1 solution

Ivan Koswara
May 25, 2015

The probability of passing is equal to the probability of answering at least $15$ questions correctly; this is:

$\begin{aligned}\displaystyle &\sum_{k=15}^{20} \binom{20}{k} \left( \frac{1}{2} \right)^k \left( \frac{1}{2} \right)^{20-k} \\ =\, &\displaystyle\frac{1}{2^{20}} \sum_{k=15}^{20} \binom{20}{k} \\ =\, &\dfrac{21700}{1048576} \end{aligned}$

Since tests are independent (the questions are changed between tests), the expected number of trials required to get one pass is simply the inverse of this: $\boxed{\dfrac{1048576}{21700} \approx 48.32}$ .

Since tests are independent (the questions are changed between tests), the expected number of trials required to get one pass is simply the inverse of this:

I couldn't understand. Can you elaborate?

Abhishek Sharma - 5 years, 11 months ago

Can u pls elaborate ur solution?

Ashley Shamidha - 5 years, 11 months ago

True/False Test

1 solution

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