In a box we have $100$ indistinguishable coins. Each coin has a $1$ on one side and a $0$ on the other side. $20$ percent of the coins are not biased, but the rest are biased, where side $1$ has a $70$ percent probability of ocurring.
a.) If we randomly grabbed a coin from the box and threw the chosen coin just once what is the probability to get a $1$? Also, what is the conditional probability that the coin we threw was biased?
In a classroom of 10 we know that 4 students cheated.
a) If we randomly choose 4 students, what is the probability that we get exactly the 4 students that cheated. Also, what will be the probability of exactly 3 students who have cheated.
Attempt at solution:
- For the first part of the first question, I know that $20\%$ of the coins are not biased so they have a $50\%$ of $0$ or $1$ occurring on a toss. But then how can I calculate the probability of getting a 1? Will it just be $[.20 \dot\ \frac{1}{2} + .7]$? For the second part of the question, since they are asking the probability of choosing a bias coin then can I use binomial distribution to calculate that probability? Thus,
$\begin{pmatrix} 100\\1 \end{pmatrix}.7^1\left(1-.7\right)^{100-1}$.
- For the second question, I know that in total their are $10$ students. So the probability of choosing the 4 that cheated is a binomial distribution. Thus, I have that
$\begin{pmatrix} 10\\4 \end{pmatrix}.4^4(1 -.4)^{10-4}$