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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?

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  • $\begingroup$ You don't multiply by 4, you take it to the power of 4 $\endgroup$ – Jonathan Davidson Jun 25 '17 at 3:09
  • $\begingroup$ Can you explain why? $\endgroup$ – K Split X Jun 25 '17 at 3:10
  • $\begingroup$ Sure, whats your math background? $\endgroup$ – Jonathan Davidson Jun 25 '17 at 3:22
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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.

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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.

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  • $\begingroup$ Here is a tutorial on how to typeset mathematics on this site. $\endgroup$ – N. F. Taussig Jun 25 '17 at 3:53

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