Final answer:
To calculate the probability of drawing exactly 2 red marbles, calculate the number of ways to draw exactly 2 red marbles divided by the total number of possible outcomes. To calculate the probability of drawing at least 1 white marble, use the complement rule. To calculate the probability of drawing all blue marbles, calculate the number of ways to draw all blue marbles divided by the total number of possible outcomes. To calculate the probability of drawing no red marbles, use the complement rule.
Step-by-step explanation:
To calculate the probability of drawing exactly 2 red marbles, first find the total number of possible outcomes. There are 9 red marbles, 6 white marbles, and 7 blue marbles, so the total number of marbles is 9 + 6 + 7 = 22. Since we are drawing 4 marbles without replacement, the total number of possible outcomes is 22 choose 4, which can be calculated as 22! / (4! * (22-4)!). The number of ways to draw exactly 2 red marbles is 9 choose 2, which can be calculated as 9! / (2! * (9-2)!). The probability of drawing exactly 2 red marbles is then the number of ways to draw exactly 2 red marbles divided by the total number of possible outcomes.
To calculate the probability of drawing at least 1 white marble, we can use the complement rule. The complement of drawing at least 1 white marble is drawing no white marbles, so we can calculate the probability of drawing no white marbles and subtract it from 1. The number of ways to draw no white marbles is (9 choose 4) + (7 choose 4), which can be calculated as (9! / (4! * (9-4)!)) + (7! / (4! * (7-4)!)). The probability of drawing at least 1 white marble is 1 minus the probability of drawing no white marbles.
To calculate the probability of drawing all blue marbles, we can use the same logic as in part a. The number of ways to draw all blue marbles is 7 choose 4, which can be calculated as 7! / (4! * (7-4)!). The probability of drawing all blue marbles is then the number of ways to draw all blue marbles divided by the total number of possible outcomes.
To calculate the probability of drawing no red marbles, we can again use the complement rule. The complement of drawing no red marbles is drawing at least 1 red marble. We have already calculated the probability of drawing at least 1 white marble in part b, so we can use that result and subtract it from 1 to get the probability of drawing no red marbles.