Using a standard 52-card deck, find the following probabilities. Leave your answer as a reduced fraction. If a card is randomly selected, what is the probability that the card is any one suit, e.g., a diamond, heart, spade, or club? If two cards are randomly selected, what is the probability of drawing a face card first, followed by drawing a numbered card? If two cards are randomly selected, what is the probability of drawing any one suit first, followed by drawing a face card?

1. The probability of drawing any one suit:

A standard deck of 52 cards is divided into 4 suits: diamonds, hearts, spades, and clubs. So, there are 13 cards of each suit. If a card is randomly selected, the probability that the card is any one suit is the number of ways to draw one card of that suit divided by the total number of cards.

So, the probability is 13/52 = 1/4.

2. The probability of drawing a face card first, followed by a numbered card:

There are 12 face cards in a deck (4 Jacks, 4 Queens, and 4 Kings). The probability of drawing a face card first is 12/52 = 3/13.

After drawing a face card, there are 51 cards left. There are 36 numbered cards in a deck (4 each of numbers 2 through 10). So, the probability of drawing a numbered card next is 36/51 = 12/17.

The probability of both events happening is the product of their individual probabilities. So, the probability of drawing a face card first, followed by a numbered card is (3/13) * (12/17) = 36/221.

3. The probability of drawing any one suit first, followed by a face card:

The probability of drawing any one suit first is 1/4, as calculated above.

After drawing a card of one suit, there are 51 cards left. There are still 12 face cards left. So, the probability of drawing a face card next is 12/51 = 4/17.

The probability of both events happening is the product of their individual probabilities. So, the probability of drawing any one suit first, followed by a face card is (1/4) * (4/17) = 1/17.