A bag contains 9 red marbles, 10 white, 5 blue. If six are drawn at random, what is the probability of
A) 3 blue
B) same number of each colour
C) none will be red
D) all 6 will be the same colour
A) To find the probability of drawing 3 blue marbles, we need to consider the total possible outcomes and the favorable outcomes.
Total possible outcomes: Choosing 6 marbles out of 24 (9 red + 10 white + 5 blue) is given by the combination formula: C(24, 6) = 134,596.
Favorable outcomes: Choosing 3 blue marbles out of 5 is given by the combination formula: C(5, 3) = 10.
Therefore, the probability of drawing 3 blue marbles is 10/134,596.
B) To find the probability of drawing the same number of each color, we need to consider the total possible outcomes and the favorable outcomes.
Total possible outcomes: Choosing 6 marbles out of 24 (9 red + 10 white + 5 blue) is given by the combination formula: C(24, 6) = 134,596.
Favorable outcomes: There are two ways to have the same number of each color:
1. Choosing 2 blue, 2 red, and 2 white marbles: C(5, 2) * C(9, 2) * C(10, 2) = 4,050.
2. Choosing 4 blue, 1 red, and 1 white marble: C(5, 4) * C(9, 1) * C(10, 1) = 450.
Therefore, the probability of having the same number of each color is (4,050 + 450) / 134,596.
C) To find the probability of none of the drawn marbles being red, we need to consider the total possible outcomes and the favorable outcomes.
Total possible outcomes: Choosing 6 marbles out of 24 (9 red + 10 white + 5 blue) is given by the combination formula: C(24, 6) = 134,596.
Favorable outcomes: Choosing 6 marbles without any red marbles: C(15, 6) = 5005.
Therefore, the probability of none of the drawn marbles being red is 5005 / 134,596.
D) To find the probability of all 6 marbles being the same color, we need to consider the total possible outcomes and the favorable outcomes.
Total possible outcomes: Choosing 6 marbles out of 24 (9 red + 10 white + 5 blue) is given by the combination formula: C(24, 6) = 134,596.
Favorable outcomes: There are three ways to choose all 6 marbles of the same color: C(9, 6) + C(10, 6) + C(5, 6) = 84.
Therefore, the probability of all 6 marbles being the same color is 84 / 134,596.
To calculate the probability of each event, we first need to find the total number of possible outcomes.
There are a total of 24 marbles in the bag (9 red + 10 white + 5 blue). Let's calculate the number of possible outcomes for each event.
A) Probability of 3 blue marbles:
To calculate this, we need to choose 3 blue marbles out of the 5 available. The remaining 3 marbles can be of any color.
Possible outcomes = (number of ways to choose 3 blue marbles) * (number of ways to choose 3 marbles out of 24)
= (5C3) * (24C3)
B) Probability of the same number of each color:
For this event, we need to choose 2 red, 2 white, and 2 blue marbles out of their respective available quantities.
Possible outcomes = (number of ways to choose 2 red marbles) * (number of ways to choose 2 white marbles) * (number of ways to choose 2 blue marbles)
= (9C2) * (10C2) * (5C2)
C) Probability of none being red:
For this event, we need to choose 6 marbles out of the total available (excluding the 9 red marbles).
Possible outcomes = (number of ways to choose 6 marbles from the remaining marbles) / (number of ways to choose 6 marbles out of 24)
= (15C6) / (24C6)
D) Probability of all 6 marbles being the same color:
We have 3 options for the color: red, white, or blue.
For each color, we need to choose 6 marbles out of the available quantity for that color.
Possible outcomes = (number of ways to choose 6 marbles from red) + (number of ways to choose 6 marbles from white) + (number of ways to choose 6 marbles from blue)
= (9C6) + (10C6) + (5C6)
Now let's calculate the probabilities:
A) 3 blue: Probability = (number of possible outcomes for 3 blue marbles) / (total number of possible outcomes)
B) Same number of each color: Probability = (number of possible outcomes for same number of each color) / (total number of possible outcomes)
C) None will be red: Probability = (number of possible outcomes for no red marbles) / (total number of possible outcomes)
D) All 6 will be the same color: Probability = (number of possible outcomes for all 6 marbles of the same color) / (total number of possible outcomes)
Let me calculate the values for you.