A bag contains 7 red marbles, 5 white marbles, and 7 blue marbles. You draw 3 marbles out at random, without replacement. What is the probability that all the marbles are red? The probability that all the marbles are red is What is the probability that exactly two of the marbles are red? The probability that exactly two of the marbles are red is What is the probability that none of the marbles are red? The probability of picking no red marbles is

The total number of marbles is \(7 + 5 + 7 = 19\). When drawing 3 marbles without replacement, the total number of outcomes is \[ \binom{19}{3}. \] ### 1. All marbles are red There are 7 red marbles, and we want to choose 3 of them. The number of favorable outcomes is \[ \binom{7}{3}. \] Thus, the probability that all the marbles are red is \[ \frac{\binom{7}{3}}{\binom{19}{3}}. \] ### 2. Exactly two of the marbles are red Here we want 2 red marbles and 1 marble that is not red. There are 7 red marbles and \(5 + 7 = 12\) non-red marbles. - The number of ways to choose 2 red marbles is \(\binom{7}{2}\). - The number of ways to choose 1 non-red marble is \(\binom{12}{1}\). So the number of favorable outcomes is \[ \binom{7}{2} \cdot \binom{12}{1}. \] Thus, the probability that exactly two of the marbles are red is \[ \frac{\binom{7}{2} \cdot \binom{12}{1}}{\binom{19}{3}}. \] ### 3. None of the marbles are red If no red marble is drawn, then all 3 marbles must come from the 12 non-red marbles. The number of favorable outcomes is \[ \binom{12}{3}. \] Thus, the probability that none of the marbles are red is \[ \frac{\binom{12}{3}}{\binom{19}{3}}. \]