A bag contains 5 red marbles, 6 white marbles, and 8 blue marbles. You draw 5 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 ?
prob(all 5 are red)
= (5/19)(4/18)(3/17)(2/16)(1/15)
= 1/11628
or , if you know how to calculate combinations
= C(5,5)/C(19,5)
= 1/11628
from here we can do
prob(no red) = 1 - 1/11628 = 11627/11628
prob(exactly 2 are red)
= C(5,2)xC(14,3)/C(19,5)
= 10(364)/11628
= 91/2907
To calculate the probabilities, we need to consider the total number of possible outcomes and the number of favorable outcomes.
1) Probability that all marbles are red:
Total number of marbles = 5 (red) + 6 (white) + 8 (blue) = 19
Number of favorable outcomes = 5 (red marbles)
P(all marbles are red) = favorable outcomes / total outcomes = 5/19.
2) Probability that exactly two of the marbles are red:
To calculate this, we need to choose 2 red marbles and 3 marbles from the remaining white and blue marbles.
Number of ways to choose 2 red marbles out of 5 = combination(5, 2) = 5! / (2! * (5-2)!) = 10
Number of ways to choose 3 marbles out of the remaining 14 (6 white + 8 blue) = combination(14, 3) = 14! / (3! * (14-3)!) = 364
Total number of favorable outcomes = number of ways to choose 2 red marbles * number of ways to choose 3 marbles from the remaining
= 10 * 364 = 3,640
Total number of outcomes = combination(19, 5) = 19! / (5! * (19-5)!) = 116,280
P(exactly two of the marbles are red) = favorable outcomes / total outcomes = 3,640 / 116,280.
3) Probability that none of the marbles are red:
Total number of marbles = 5 (red) + 6 (white) + 8 (blue) = 19
Number of favorable outcomes = 6 (white marbles) + 8 (blue marbles) = 14
P(none of the marbles are red) = favorable outcomes / total outcomes = 14/19.
To solve these probability questions, we need to understand the concept of probability and how it is calculated.
The probability of an event occurring is calculated by dividing the number of favorable outcomes by the total number of possible outcomes.
1. Probability that all the marbles are red:
To calculate this probability, we first need to find the total number of possible outcomes. In this case, there are 5 red marbles out of a total of 19 marbles (5 red + 6 white + 8 blue). When we draw marbles without replacement, the total number of possible outcomes decreases with each draw.
Probability = (Number of favorable outcomes) / (Total number of possible outcomes)
= (Number of ways to select 5 red marbles out of 5) / (Number of ways to select 5 marbles out of 19)
= 1 / (19 C 5)
= 1 / 1,277
Therefore, the probability that all the marbles are red is 1/1,277.
2. Probability that exactly two of the marbles are red:
To calculate this probability, we need to determine the number of ways to select exactly two red marbles out of five. We also need to consider the outcomes for the remaining three marbles, which will not be red.
Probability = (Number of favorable outcomes) / (Total number of possible outcomes)
= (Number of ways to select 2 red marbles out of 5) * (Number of ways to select 3 non-red marbles out of 14) / (Number of ways to select 5 marbles out of 19)
= (5 C 2) * (14 C 3) / (19 C 5)
= 10 * 364 / 1,277
Therefore, the probability that exactly two marbles are red is approximately 28.6% (364/1,277).
3. Probability that none of the marbles are red:
In this case, we want to find the number of ways to select all non-red marbles (6 white and 8 blue) out of 19 total marbles.
Probability = (Number of favorable outcomes) / (Total number of possible outcomes)
= (Number of ways to select 5 non-red marbles out of 14) / (Number of ways to select 5 marbles out of 19)
= (14 C 5) / (19 C 5)
= 2,002 / 1,277
Therefore, the probability that none of the marbles are red is approximately 156.8% (2,002/1,277).