### Instruction: Two cards are chosen at random from a standard 52-card deck. What is the probability that they are either both hearts or both diamonds?### Response: I need to find the probability of two events: both hearts or both diamonds.
These are mutually exclusive events, so I can add their probabilities.
To find the probability of both hearts, I need to multiply the probability of the first card being a heart by the probability of the second card being a heart, given that the first card was a heart.
The probability of the first card being a heart is 13/52, since there are 13 hearts in the deck and 52 cards total.
The probability of the second card being a heart, given that the first card was a heart, is 12/51, since there are 12 hearts left in the deck and 51 cards total.
So the probability of both hearts is 13/52 * 12/51, which simplifies to 1/17.
To find the probability of both diamonds, I can use the same logic as for both hearts.
The probability of the first card being a diamond is 13/52, and the probability of the second card being a diamond, given that the first card was a diamond, is 12/51.
So the probability of both diamonds is also 13/52 * 12/51, which is also 1/17.
Therefore, the probability of either both hearts or both diamonds is 1/17 + 1/17, which is 2/17.