Can someone please explain how to solve this problem?
A bag contains 6 blue marbles, 10 red marbles, and 9 green marbles. If two marbles are drawn at random without replacement, what is the probability that two green marbles are drawn?
total marbles = 6 + 10 + 9 = 25
first draw 9/25
second draw 8/24
so
(9/25)(1/3) = 9/75
To solve the problem, we need to find the probability of drawing two green marbles out of the total number of possible outcomes.
Step 1: Find the total number of marbles in the bag.
The bag contains 6 blue marbles, 10 red marbles, and 9 green marbles. So the total number of marbles is:
Total marbles = 6 + 10 + 9 = 25
Step 2: Find the number of ways to select two green marbles.
To select two green marbles, we need to choose 2 marbles from the 9 green marbles. This can be calculated using the combination formula, denoted by "C" or "nCr," which represents the number of ways to choose r objects from a set of n objects without replacement.
In this case, n = 9 and r = 2. The number of ways to select two green marbles is:
Number of ways to select two green marbles = 9C2 = (9! / (2! * (9-2)!)) = (9! / (2! * 7!)) = (9 * 8) / (2 * 1) = 36
Step 3: Find the total number of ways to select any two marbles.
The total number of ways to select any two marbles can be calculated using the combination formula, considering all the marbles in the bag. In this case, n = 25 (total marbles) and r = 2 (number of marbles to be selected). The total number of ways to select any two marbles is:
Total number of ways to select two marbles from 25 = 25C2 = (25! / (2! * (25-2)!)) = (25! / (2! * 23!)) = (25 * 24) / (2 * 1) = 300
Step 4: Calculate the probability of drawing two green marbles.
The probability of an event is the number of favorable outcomes divided by the number of possible outcomes. In this case, the number of favorable outcomes (green marbles) is 36, and the number of possible outcomes (any two marbles) is 300. Thus, the probability of drawing two green marbles is:
Probability of drawing two green marbles = Number of ways to select two green marbles / Total number of ways to select two marbles from 25
Probability of drawing two green marbles = 36 / 300 = 0.12 or 12%
Therefore, the probability of drawing two green marbles is 0.12 or 12%.