Question

2. Two dice are thrown. What is the conditional probability of a total score of at least 8, given that one of the dice has thrown 4 ?

3. What are the chances of seeing 11 or more heads in a trial of 15 unbiased coin throws? Would you reject the hypothesis that the coin is unbiased if you see 11 heads in 15 throws?

4. What is the probability of having the 64th head on the 99th toss?

Answer 1

Problem # 6167 Two dice are thrown. What is the conditional probability of a total score of at least 8, given that one of the dice has thrown 4? Please explain your answer. Solution We will use the classical definition of probability. The space of elementary events is $\Omega = \{(i,j)|1\leq i,j\leq 6\}$, $|\Omega| = 36$. Denote by $A =$ (a total score is at least 8), $B =$ (one of the dice is 4) $= \{(i,4),(4,i)|1\leq i\leq 6\}$. Then, $|B| = 2\cdot 6 = 12$ and $\mathbb{P}\{B\} = \frac{12}{36} = \frac{1}{3}$, $|A\cap B| = 2\cdot 3 = 6$. We are interested in $\mathbb{P}\{A|B\} = \frac{\mathbb{P}\{A\cap B\}}{\mathbb{P}\{B\}} = \frac{6 / 36}{1 / 3} = 1 / 2$. Answer. 1/2.