# Probability of getting all four answers right, with true and false in each question

Suppose there are 4 questions in total.

Each question has a true/false option.

1) TRUE or FALSE

2) TRUE or FALSE

3) TRUE or FALSE

4) TRUE or FALSE

What is the probability of you getting all four questions correct?

I am just curious behind the math, I have not taken a probabiltity course so I'm sure my answer is wrong:

I think that since there are two options per question, the chances of getting 1 question correct is $50/100$. There are four questions in total, so I believe $0.5 \times 4 = 2 = 200\%$

This is obviously false. What is the right answer, and why?

• You don't multiply by 4, you take it to the power of 4 Jun 25, 2017 at 3:09
• Can you explain why? Jun 25, 2017 at 3:10
• Sure, whats your math background? Jun 25, 2017 at 3:22

For each of the four questions, there are two possible outcomes. Thus, in total, there are $2^4 = 16$ ways to answer the four questions.
\begin{array}{c c c c} T & T & T & T\\ T & F & T & T\\ T & T & F & T\\ T & F & F & T\\ T & T & T & F\\ T & F & T & F\\ T & T & F & F\\ T & F & F & F\\ F & T & T & T\\ F & F & T & T\\ F & T & F & T\\ F & F & F & T\\ F & T & T & F\\ F & F & T & F\\ F & T & F & F\\ F & F & F & F \end{array}

Only one of these $16$ sequences is correct. Assuming each of these $16$ sequences is equally likely to occur (as would result from random guessing), the probability that all four questions are answered correctly is $1/16$.

If we assume that a person is equally likely to guess true or false on each question, then he or she has probability $1/2$ of answering each question correctly. Under the assumption of independence, the probability that all four questions are answered correctly is $$\left(\frac{1}{2}\right)^4 = \frac{1}{16}$$ We add probabilities of mutually exclusive events (events that cannot occur at the same time). We multiply probabilities of independent events.

So the most direct way of telling you what the answer is is this:

For each question you have 1/2 chance of getting the right answer.

Since there are four questions, each has a 1/2 chance of getting the right answer. So to find it, you do (1/2)x(1/2)x(1/2)x(1/2). Therefore, for this case, there is a 1/16 chance of getting the right answer or (1/(2^4)) chance. Or a 6.25% chance.

• Here is a tutorial on how to typeset mathematics on this site. Jun 25, 2017 at 3:53