An urn contains N red chips, N blue chips, and N white chips all numbered 1 through N. Two chips are drawn at random and without replacement. Let A be the event that the two chips drawn are the same color; and let B be the event that they have the same number.
a) Find $P(A)$
b) Find $P(B)$
C) Find the probability that the two chips drawn are either the same color or the same number.
Here is what I have so far, but I am not sure as to whether or not it is correct.
ANSWER:
a) $P(A)= \binom{R}{2} + \binom{B}{2} \binom{W}{2}$
b) $P(B)= \frac{3}{R+B+W} \times \frac{2}{R+B+W-1}$
c) $P(\text{same color or same number})= P(A)+P(B)$
Any help that can be offered with a detailed explanation as to how I would be able to relate it to future problems would be amazing!
Best Answer
For the first problem we now that there $\binom{N}{2}$ to choose a pair of red chips, also this holds for the the blue chips and white chips. Also there are $\binom{3N}{2}$ to ways to draw two chips, because there are $3N$ chips in the urn. So the probability is:
$$P(A) = \frac{3\binom{N}{2}}{\binom{3N}{2}} = \frac{\frac{3N(N-1)}{2}}{\frac{3N(3N-1)}{2}} = \frac{N-1}{3N-1}$$
For the second case there are 3 balls with same number at the beginning, while for the second draw there are 2 balls with the same number as the one we draw earlier. Note that the first draw doesn't matter, because we only need to draw the same numbered ball as the one we draw earlier. So the probability is:
$$P(B) = N \times \frac{3}{N+N+N} \times \frac{2}{N+N+N-1}$$ $$P(B) = \frac{2}{3N-1}$$
And for the third case we need to check when exactly one of the conditions is fulfiled, we are using the exclusive OR. So using inclusion-exclusion we have:
$$P(A \oplus B) = P(A) + P(B) - P(A \cup B)$$
But note that the event $A$ and event $B$ never intersect, because it's impossible to draw two balls from the same colour and with same number same number, so we have:
$$P(A \oplus B) = P(A) + P(B) - P(A \cup B)$$ $$P(A \oplus B) = \frac{N-1}{3N-1} + \frac{2}{3N - 1}$$ $$P(A \oplus B) = \frac{N+1}{3N-1}$$