SOLUTION: Run has 7 red marbles, 7 blue marbles and 6 white marbles. I take 3 marbles with no replacement. What is the probability that all 3 are the same color? What is the probability t

Question 1164015: Run has 7 red marbles, 7 blue marbles and 6 white marbles. I take 3 marbles with no replacement. What is the probability that all 3 are the same color? What is the probability that none are white?
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!

Part A) The probability all three are the same color

7 red + 7 blue + 6 white = 20 total
The probability of getting one red is 7/20
The probability of getting another red 6/19 since we don't replace the first marble we picked
The probability of getting a third red 5/18
You effectively count down for the numerator and denominator

You then multiply all the fractions like so
P(3 red) = probability of getting 3 red marbles in a row, no replacement
P(3 red) = P(1st red)*P(2nd red)*P(3rd red)
P(3 red) = (7/20)*(6/19)*(5/18)
P(3 red) = (7*6*5)/(20*19*18)
P(3 red) = 210/6840
P(3 red) = (7*30)/(228*30)
P(3 red) = 7/228

We'll apply the same idea for the blue ones as well
P(3 blue) = P(1st blue)*P(2nd blue)*P(3rd blue)
P(3 blue) = (7/20)*(6/19)*(5/18)
P(3 blue) = 7/228
We get the same value as before because the number of red marbles equals the number of blue marbles.

The white marbles is a slightly different story
P(3 white) = P(1st white)*P(2nd white)*P(3rd white)
P(3 white) = (6/20)*(5/19)*(4/18)
P(3 white) = (6*5*4)/(20*19*18)
P(3 white) = 120/6840
P(3 white) = (1*120)/(57*120)
P(3 white) = 1/57

Now add all of the probabilities found to get the answer we're after
P(3 same color) = P(3 red) + P(3 blue) + P(3 white)
P(3 same color) = 7/228 + 7/228 + 1/57
P(3 same color) = 7/228 + 7/228 + 4/228
P(3 same color) = (7+7+4)/228
P(3 same color) = 18/228
P(3 same color) = (3*6)/(38*6)
P(3 same color) = 3/38
Note how the events "3 red", "3 blue" and "3 white" are mutually exclusive

Final Answer: 3/38

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Part B) The probability none of the three are white

The event "none of the 3 are white" is the same as "the three marbles are either red or blue only"

We have 7 red+7 blue = 14 marbles that are either red or blue.
This is out of 20 total

P(none are white) = P(3 red or blue)
P(none are white) = P(1st red or blue)*P(2nd red or blue)*P(3rd red or blue)
P(none are white) = (14/20)*(13/19)*(12/18)
P(none are white) = 91/285

Final Answer: 91/285

Side note: It might be tempting to say the answer is 1-P(3 white), but this would not be correct. This computes the probability that at most 2 are white (ie, 0 white, 1 white or 2 white). It is the complement of getting 3 white marbles in a row.

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