I have 5 red plates and 4 blue plates. If I randomly select two plates to serve dinner on, what is the probability that they're both the same color?
There are ${9 \choose 2} = 36$ pairs of plates possible in all. Exactly ${5 \choose 2} = 10$ of these pairs are both red plates and $\binom{4}{2}=6$ pairs are both blue plates for a total of $6+10=16$ pairs of plates that satisfy our condition. So, the probability of the pair of plates being the same color is $\frac{16}{36}=\boxed{\frac{4}{9}}$.
\frac{4}{9}
Level 3
Counting & Probability
Problem: I have 5 red plates and 4 blue plates. If I randomly select two plates to serve dinner on, what is the probability that they're both the same color?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the probability that both plates selected are the same color, we first determine the total number of plates. There are 5 red plates and 4 blue plates, giving a total of 9 plates. 

Next, we calculate the total number of ways to select 2 plates from these 9 plates. This can be done using the combination formula, which is given by C(n, k) = n! / (k!(n-k)!), where n is the total number of items to choose from, and k is the number of items to choose. 

Thus, the total number of ways to choose 2 plates from 9 is:
C(9, 2) = 9! / (2!(9-2)!) = 9! / (2!7!) = (9 × 8) / (2 × 1) = 36.

Now, we need to find the number of favorable outcomes where both plates are of the same color. 

First, we calculate the number of ways to select 2 red plates from the 5 red plates:
C(5, 2) = 5! / (2!(5-2)!) = 5! / (2!3!) = (5 × 4) / (2 × 1) = 10.

Next, we calculate the number of ways to select 2 blue plates from the 4 blue plates:
C(4, 2) = 4! / (2!(4-2)!) = 4! / (2!2!) = (4 × 3) / (2 × 1) = 6.

Now, we add the number of favorable outcomes:
Total favorable outcomes = Number of ways to choose 2 red plates + Number of ways to choose 2 blue plates = 10 + 6 = 16.

Finally, we calculate the probability that both plates selected are of the same color by dividing the number of favorable outcomes by the total number of outcomes:
Probability = Total favorable outcomes / Total outcomes = 16 / 36.

To simplify this fraction, we divide both the numerator and the denominator by their greatest common divisor, which is 4:
16 / 36 = 4 / 9.

Thus, the final answer is 4/9.<|END_OF_LATENT|> The final answer is $\frac{4}{9}$.