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# Probability

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Raina places the following balls into a bag.  She draws three balls out of the bag, one at a time, without replacement.  What is the probability that the colors of the balls alternate?

There are 6 orange balls and 6 purple balls.

Dec 24, 2023

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Here's how to find the probability of Raina drawing three balls with alternating colors:

Method 1: Counting cases

Favorable cases: There are two ways the colors can alternate: orange-purple-orange (OPO) or purple-orange-purple (POP).

For OPO: Choose an orange ball first (6 options), then a purple ball (6 options), and then another orange ball (5 remaining options). This gives 6 * 6 * 5 = 180 ways.

For POP: Choose a purple ball first (6 options), then an orange ball (6 options), and then another purple ball (5 remaining options). This gives 6 * 6 * 5 = 180 ways.

Total favorable cases: 180 + 180 = 360.

Total cases: Choose any 3 balls from the 12 total balls (6 orange, 6 purple). This gives 12C3 = 220 ways.

Probability: Divide the number of favorable cases by the total number of cases: 360 / 220 = 18/11.

Method 2: Using complementary probability

Calculate the probability of not drawing three balls with alternating colors:

This means either drawing all orange balls (OOO) or all purple balls (PPP).

Probability of OOO: (6/12) * (5/11) * (4/10) = 60/660 = 3/33

Probability of PPP: (6/12) * (5/11) * (4/10) = 60/660 = 3/33

Total probability of not alternating: 3/33 + 3/33 = 6/33

Subtract the probability of not alternating from 1 to get the probability of alternating: 1 - 6/33 = 18/11.

Therefore, the probability of Raina drawing three balls with alternating colors is 18/11.

Dec 24, 2023