From a standard deck of 52 cards, all of the clubs and the jack of diamonds are removed. Two cards are drawn at random from the altered deck without replacement. What is the probability that both cards will be of the same suit? Justify your answer, which should be a lowest terms fraction.
To find the probability that both cards will be of the same suit, we need to first determine the number of favorable outcomes (i.e., the number of ways we can choose two cards of the same suit) and the number of possible outcomes (i.e., the total number of ways we can choose two cards from the altered deck).
Let's start by calculating the number of favorable outcomes. Since all the clubs and the jack of diamonds have been removed, we are left with 39 cards in the deck. Each suit now has 13 cards remaining (hearts, diamonds, and spades). So we have three suits to choose from.
To calculate the number of ways we can choose two cards of the same suit, we need to consider all the different possibilities. We can choose two cards from the hearts, two cards from the diamonds, or two cards from the spades.
For the hearts, there are 13 cards remaining, so we have 13C2 (read as "13 choose 2") ways to choose two cards from the hearts.
Similarly, for the diamonds and spades, we also have 13C2 ways to choose two cards from each suit.
So the total number of favorable outcomes is 13C2 + 13C2 + 13C2 = 3 * 13C2.
Next, let's determine the number of possible outcomes, which is the total number of ways we can choose two cards from the altered deck. Since we have 39 cards remaining in the deck after removing the clubs and jack of diamonds, we have 39C2 ways to choose any two cards.
Now, we can calculate the probability by dividing the number of favorable outcomes by the number of possible outcomes:
Probability = (Number of favorable outcomes) / (Number of possible outcomes).
Probability = (3 * 13C2) / 39C2.
To simplify the expression, we can calculate the combinations using the formula:
nCr = n! / (r! * (n - r)!)
Using this formula, we can calculate:
13C2 = 13! / (2! * 11!) = (13 * 12) / (2 * 1) = 78.
39C2 = 39! / (2! * 37!) = (39 * 38) / (2 * 1) = 741.
So, the probability is:
Probability = (3 * 78) / 741 = 234 / 741.
To put it in lowest terms, we can simplify the fraction:
Probability = 26 / 83.
Therefore, the probability that both cards will be of the same suit is 26/83.