Three marbles are drawn without replacement from an urn containing 4 red marbles, five what marbles, and two blue marbles. Determine the probability that None is red
4 red out of (4+5+2)=11 marbles
Probability that first draw has no red
=(11-4)/11
=7
No replacement, so probability for second draw
= (10-4)/10
=6/10
Probability that both have no red
= product of the two probabilities.
To determine the probability that none of the three marbles drawn is red, we need to find the ratio of the favorable outcomes (no red marbles) to the total number of possible outcomes.
Step 1: Calculate the total number of possible outcomes.
We have a total of 11 marbles in the urn: 4 red, 5 white, and 2 blue. When drawing three marbles without replacement, the total number of possible outcomes is given by the combination formula:
Total number of possible outcomes = C(11, 3) = 11! / (3!(11-3)!) = 55.
Step 2: Calculate the number of favorable outcomes (no red marbles).
Since we want none of the three marbles to be red, we can only select from the total of 7 non-red marbles (5 white + 2 blue). Using the combination formula again:
Number of favorable outcomes = C(7, 3) = 7! / (3!(7-3)!) = 35.
Step 3: Calculate the probability.
The probability of none of the three drawn marbles being red is given by the ratio of the number of favorable outcomes to the total number of possible outcomes:
Probability = Number of favorable outcomes / Total number of possible outcomes
= 35 / 55
= 7 / 11
≈ 0.6364
Therefore, the probability that none of the three marbles drawn is red is approximately 0.6364 or 63.64%.
To determine the probability that none of the three marbles drawn is red, we need to find the number of ways to choose three marbles from the urn that are not red, and divide it by the total number of ways to choose three marbles.
First, let's calculate the total number of ways to choose three marbles from the urn. We have a total of 4 red marbles, 5 white marbles, and 2 blue marbles, which gives us a total of 11 marbles.
The number of ways to choose three marbles out of 11 is given by the combination formula:
C(n, r) = n! / (r! * (n-r)!)
where n is the total number of marbles and r is the number of marbles we want to choose.
Using this formula, we can calculate C(11, 3) as follows:
C(11, 3) = 11! / (3! * (11-3)!)
= 11! / (3! * 8!)
= (11 * 10 * 9) / (3 * 2 * 1)
= 165
So, there are 165 ways to choose three marbles from the urn.
Next, let's calculate the number of ways to choose three marbles that are not red. Since we do not want any red marbles, we need to choose three marbles from the remaining 7 (5 white marbles + 2 blue marbles).
Using the combination formula again, we can calculate C(7, 3) as follows:
C(7, 3) = 7! / (3! * (7-3)!)
= (7 * 6 * 5) / (3 * 2 * 1)
= 35
So, there are 35 ways to choose three marbles from the remaining 7 marbles that are not red.
Finally, we can calculate the probability by dividing the number of favorable outcomes (choosing three marbles that are not red) by the total number of possible outcomes (choosing three marbles from the urn).
Probability = Number of favorable outcomes / Total number of possible outcomes
= 35 / 165
= 7 / 33
Therefore, the probability that none of the three marbles drawn is red is 7/33.