Solve the following two independent problems. (i) A mother and her two daughters participate in a game show. At first, the mother tosses a fair coin. Case 1: If the result is heads, then all three win individual prizes and the game ends. Case 2: If the result is tails, then each daughter separately throws a fair die and wins a prize if the result of her die is 5 or 6. (Note that in case 2 there are two independent throws involved and whether each daughter gets a prize or not is unaffected by the other daughter's throw.) (a) Suppose the first daughter did not win a prize. What is the probability that the second daughter also did not win a prize? (b) Suppose the first daughter won a prize. What is the probability that the second daughter also won a prize? (ii) Prove or disprove each of the following statements. (a) $2 ^ { 40 } > 20!$ (b) $1 - \frac { 1 } { x } \leq \ln x \leq x - 1$ for all $x > 0$.
Solve the following two independent problems.
(i) A mother and her two daughters participate in a game show. At first, the mother tosses a fair coin.
Case 1: If the result is heads, then all three win individual prizes and the game ends.\\
Case 2: If the result is tails, then each daughter separately throws a fair die and wins a prize if the result of her die is 5 or 6. (Note that in case 2 there are two independent throws involved and whether each daughter gets a prize or not is unaffected by the other daughter's throw.)
(a) Suppose the first daughter did not win a prize. What is the probability that the second daughter also did not win a prize?\\
(b) Suppose the first daughter won a prize. What is the probability that the second daughter also won a prize?
(ii) Prove or disprove each of the following statements.\\
(a) $2 ^ { 40 } > 20!$\\
(b) $1 - \frac { 1 } { x } \leq \ln x \leq x - 1$ for all $x > 0$.