What is the probability that the two balls have different colors?

Question

There are two urns: the one with 2 white and 3 black balls and the other one with 3 white and 3 black balls. We pick one urn at random and draw two balls out of it. What is the probability that the two balls have different colors?

I have tried like the following…

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let, two urns are X and Y.

$X = \{W,W,B,B,B\}$

$Y = \{W,W,W,B,B,B\}$

Probability of selecting a pair $\{B,W\}$ from $X$ is,

$\frac{^3C_1 \cdot ^2C_1}{^5C_2} = \frac{3}{5}$

Probability of selecting a pair $\{B,W\}$ from $Y$ is,

$\frac{^3C_1 \cdot ^3C_1}{^6C_2} = \frac{3}{5}$

Probability of picking an urn at random is $\frac{1}{2}.$

If event $A$ = picking the pair $\{B,W\}$.

Then,

$P(A) = P(X).P(A|X) + P(Y).P(A|Y)$

$\Rightarrow P(A) = \frac{3}{10} + \frac{3}{10} = \frac{3}{5}$

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Am I correct?

Answer 1

Yes, although once you realize the probability doesn't depend on which urn you draw from you can just stop there and declare that the answer.

Just for fun we can create a simulation in R to estimate the probability.

n = 100000
count = 0

for (i in 1:n) {
    # generate a random urn (0's will be white balls and 1's black balls)
    urn = c(rep(0, 2 + rbinom(n=1, size=1, prob=1/2)), rep(1, 3))
    # draw two elements from the urn and check if their sum is one
    count = count + as.numeric(sum(sample(urn, 2, replace=FALSE)) == 1)
}

# look at the proportion of successes
count / n

If you run this you should get an answer very close to $3/5$.