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?
Best Answer
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.