Given a bag containing 10 red balls and 10 green balls. When you draw 6 balls, without replacement, what is the probability that you will have at least 1 red and 1 green ball.

I attempted this initially by figuring about the probability of drawing at least 1 red, which is 1 - P(no red). (http://mathforum.org/library/drmath/view/69151.html) Then I just figured that P(>=1 red) * P(>=1 green) = P(>= 1 red & >= 1 green) but since they aren't independent. (i.e. You can't draw 6 red & 6 green when drawing 6 total) The only way I could concieve a solution would be to subtract the overlapping events.

This leaves me with

P(>= 1 red & >= 1 green) = P(>= 1 red)*P(>= 1 green) - P(=6 red)*P(=6 green) - P(=6 red)*P(=5 green) - ... - P(=1 red)*P(=6 green)

Which I think is "Correct" but I feel as though there should be a more intuitive way to calculate this.


There certainly is a more intuitive way at looking at this:

Consider the inverse of this statement, where we try and calculate the probability that not at least one green and red ball are chosen. Therefore, all the balls chosen will be either green or red.

So, for all green:

$$P =\frac{10}{20}*\frac{9}{19}*\frac{8}{18}*\frac{7}{17}*\frac{6}{16}*\frac{5}{15} = \frac{7}{1292}$$

Since you have the same number of red balls, it will be the same probability.

Therefore, the total probability is: $$\frac{7}{1292}*2 = \frac{14}{1292}$$

Now we simply subtract this from the total probability, $1$:

$$1 - \frac{14}{1292} = \frac{639}{646}$$

Comment if you have any questions.

  • $\begingroup$ How would the problem change if we instead had 3 colors and you wanted at least 1 of each, lets say 10 red, 10 green, and 10 white? $\endgroup$ – Rovert Renchirk Aug 15 '14 at 4:02
  • $\begingroup$ You would do something similarly where you subtract the probabilities of getting all white all green and all red. $\endgroup$ – Varun Iyer Aug 15 '14 at 11:00

There are $\binom{20}{6}$ ways to select the balls, and the number of ways to select them so that you do not get at least one red and at least one green is given by $2\binom{10}{6}$,

since there are $\binom{10}{6}$ ways to select 6 red balls (or 6 green balls).

Therefore the probability is $\displaystyle1-\frac{2\binom{10}{6}}{\binom{20}{6}}=1-\frac{420}{38760}=1-\frac{7}{646}=\frac{639}{646}$.


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