Question

A quiz has 1 true-false question and 2 multiple choice questions that each have 4 answers. What is the probability that someone guessing all three answers will get \[100\% \] in the quiz?

Answer 1

Hint: We will use the concepts of probability of events to solve this problem.
We use the formula \[P(E) = \dfrac{{{\text{number of favorable outcomes}}}}{{{\text{total number of outcomes}}{\text{.}}}}\] which gives us the probability of an event to happen.

Complete step by step answer:
In mathematics, probability is defined as the occurrence of a random event. It is also defined as the ratio of number of favorable outcomes to total number of outcomes.
So, \[P(E) = \dfrac{{{\text{number of favorable outcomes}}}}{{{\text{total number of outcomes}}{\text{.}}}}\] is the probability of an event E.
If the probability of an event is 0, then the event doesn’t happen.
If the probability of an event is 1, then it will happen for sure.
So, in a true-false question, there are only two options i.e., either true or false and one of them is correct.
So, the probability of guessing correctly is \[\dfrac{1}{2}\].
And in a multiple-choice question, there are four options out of which only one is correct.
So, the probability of guessing a correct option is \[\dfrac{1}{4}\].
Now, there are a total of three questions i.e., one true-false and two multiple choice questions.
Let the event of guessing all the questions correct be C.
So, probability of guessing all three questions correctly is \[P(C) = \dfrac{1}{2} \times \dfrac{1}{4} \times \dfrac{1}{4}\]
\[ \Rightarrow P(C) = \dfrac{1}{{32}}\]
Or we can also write this as
\[ \Rightarrow P(C) = 0.03125\]
So, this is the required answer.

Note:
Make a note that, probability of an event will never be a negative value or a value greater than 1. If you get such value, then you have committed a mistake in your solution. The range of probability of an event to happen is \[[0,1]\].
Probability of a total event which is having sub-events, is equal to the product of probabilities of individual sub-events.