Mick thinks that he can read people's minds. To test his hypothesis you perform an experiment. you think of a number between $1$ and $10$, and then he guesses this number. This is then repeated two more times (for a total of three numbers/guesses). If he guesses right at least twice, you accept his claim that he can read minds.
Calculate the power of your test, if Mick can read minds well enough to guess the right number $0.8$ of the time.
I know that the power of your test can be calculated by P(reject null hypothesis | null hypothesis is false) but I'm just really confused on how to calculate the probability of the null hypothesis being false and what the $0.8$ actually means in the question.
would appreciate the help
Let $X$ be the number of correct guesses in $3$ trials. You reject the null hypothesis if $X \geq 2$. Since Mick's probability of success in any given trial is $p = 0.8$, you can use the binomial distribution, which states that the probability of exactly $k$ successes in $n$ trials, each of which has probability $p$ of success, is $$\Pr(X = k) = \binom{n}{k}p^k(1 - p)^{n - k}$$ where $\binom{n}{k}$ counts the number of ways exactly $k$ successes could occur in $n$ trials, $p^k$ is the probability of $k$ successful trials, and $(1 - p)^{n - k}$ is the probability of failure in the remaining $n - k$ trials.