Question

A box contains three coins, one coin is fair, one coin is two headed and one coin is weighted so that is weighted so that the probability of head appearing is $\frac{1}{3}$. A coin is selected at random is tossed. Find the probability $p$ that head appears?

Answer 1

For part , we just need to compute $P\left(H\right)$. since $T,F$ and $W$ partition the sample space (i.e., they are pairwise disjoint and their union accounts for all possibilities), we can use the law of total probability to get

$P\left(H\right)=P\left(\frac{H}{T}\right)P\left(T\right)+P\left(\frac{H}{F}\right)P\left(F\right)+P\left(\frac{H}{W}\right)P\left(W\right)=\left(1\right)\left(\frac{1}{3}\right)+\left(\frac{1}{2}\right)\left(\frac{1}{3}\right)+\left(\frac{1}{3}\right)\left(\frac{1}{3}\right)=\frac{33}{54}$.