Drawing Cards from a Deck

Question

You have a standard deck of 52 cards. When drawing 3 cards randomly from this deck, answer the following questions:

What is the probability of drawing at least one Ace card (excluding Jokers)? What is the probability of drawing 3 cards of the same suit (one of Hearts, Diamonds, Spades, or Clubs)?

Answer 1

There are 4 Aces in a standard deck, and there are 48 non-Ace cards

The probability of not drawing an Ace in the first draw is (48/52). The probability of not drawing an Ace in the second draw, after not drawing one in the first, is (47/51). The probability of not drawing an Ace in the third draw, after not drawing one in the first and second, is (46/50).

Now, multiply these probabilities together:

$p= (48/52) \cdot (47/51) \cdot (46/50)$

This is the probability of not drawing any Aces in all three draws. To find the probability of drawing at least one Ace, subtract this from 1:

$1 - p$

Now, to calculate the probability of drawing 3 cards of the same suit (Hearts, Diamonds, Spades, or Clubs), you can use the following approach:

1. Calculate the probability of drawing 3 cards of the same suit (e.g., Hearts).
2. Multiply this probability by 4 (since there are 4 suits in a standard deck) to account for all possible suits.

So,

$p= 4\cdot (13/52) \cdot (12/51) \cdot (11/50)$

This is because there are 13 cards of each suit in a standard deck, and we’re calculating the probability of drawing one from that suit on each of the three draws.