A student is writing a test with 4 true/false questions. The student wants to know the theoretical probability of passing the test by guessing. This means that 2 or more of the 4 questions are answered correctly by guessing. Determine this probability and show all your work.
I don't know what to do next and how to approach it.
SO since there is 2 options and 4 questions it would be 0.5^2
But It says 2 or more, which is why I am confused.
Best Answer
The student can get $2$ questions right and $2$ wrong, or $3$ right and $1$ wrong, or all $4$ right. For him to get all $4$ right, the probability is $0.5^4$.
For him to get $3$ right and $1$ wrong, there are $4$ ways to pick which question he gets wrong, then $0.5$ probability that he gets it wrong, then $0.5^3$ probability he gets the other $3$ questions right.
Finally, for him to get $2$ right and $2$ wrong, there are $\binom{4}{2} = 6$ ways to pick which $2$ questions he gets right, $0.5^2$ probability he gets them right, and then $0.5^2$ probability he gets the other $2$ questions wrong.
Therefore, the total probability is $0.5^4 + 4 \cdot 0.5^3 \cdot 0.5 + 6 \cdot 0.5^2 \cdot 0.5^2 = 0.6875$.
Note: Considering this sort of problem in the general case, where $4$ is replaced by $n$, $2$ is replaced by $k$, and $0.5$ is replaced by $p$, leads to the idea of a binomial random variable (or a random variable with the binomial distribution). You can look at http://onlinestatbook.com/2/probability/binomial.html.