Question

On a true-false quiz, the probability of achieving a perfect score by guessing is halved for each additional question on the quiz. This probability equals 1/2 for a one-question quiz. What is the minimum number of questions needed for a true-false quiz, so that the probability of achieving a perfect score by guessing is less than 1/10

Answer 1

For each question, there is a 1/2 chance at getting the question correct by guessing.

Let's take a scenario to better understand.
Suppose the true-false paper has 5 questions. For a perfect score by guessing, you'd need to get all 5 correct (ie (1/2)⁵)
The reason why you multiply is because you need each 1/2 simultaneously for a perfect score, which is an important concept when doing binomial probability later on.

Thus, let's use this knowledge to answer the question.
We need the minimum amount of questions such that the probability is less than 1/10.

We can write an inequality for this:

$\frac{1}{10} < \frac{1}{2}^{n}$
Now, we need to log both sides to find n.
$log_(\frac{1}{2}) \frac{1}{10} < n$
n > 3.3219...
n ≈ 4

Thus, 4 questions is the minimum number of questions needed.