Probability of drawing 5 cards from a deck of 52 that will have the same suit?

Question

A standard deck of cards has 52 members consisting of 4 suits each with 13 members. Five cards are dealt from the randomly mixed deck. What is the probability that all cards are the same suit?

EDIT: How I went about it before posting this question was doing (1/4) as the first card probability because my thought process was that we'll draw 1 suit out of the 4 for the first probability. Then I proceeded to account of the 2nd dealt card with the probability of (12/51) since 1 card has been dealt already out of the 13 cards for that suit, also subtracting 1 from the total amount of cards able to be dealt.

So for the 3rd card: (11/50)

4th card: (10/49)

5th card: (9/48)

Giving us the total overall probability for drawing 5 cards of the same suit: $$ (1/4) * (12/51) * (11/50) * (10/49) * (9/48) = 33/66640 $$

EDIT2: My practice quiz given by TA's is still saying I have the incorrect answer. Given how the answer should be: $ 33/16660 $ (explained in numerous ways in the thread), I contacted the TA's to see if maybe the have setup the question incorrectly. Will update when I get an answer back.

EDIT3: Got an answer back from my TA's who tested the test. They did have the answer wrong on their end. Everyone who helped me was correct!

Answer 1

The first card has probability $\frac{52}{52}$ of having the same suit as any previously drawn cards (because there are none). This means there is a 100% chance of the first card meeting our criteria.

The second card has probability $\frac{12}{51}$ because there are twelve left out of 51 total that match the suit of the first card.

The third, fourth and fifth cards have probabilities $\frac{11}{50}$, $\frac{10}{49}$, and $\frac{9}{48}$ because there are less and less of the suit of the first card as well as less cards to choose from.

Because I need the first thing to happen AND the second thing to happen AND, ..., I need to multiply the probabilities: $\big(\frac{52}{52}\big)\big(\frac{12}{51}\big)\big(\frac{11}{50}\big)\big(\frac{10}{49}\big)\big(\frac{9}{48}\big)$

You obviously don't need that first fraction, but I think it adds clarity. Five cards, each with their own probability.

Answer 2

We know that the first card determines the suit, the second must be one of the $12$ out of $51$ remaining cards with that suit, the third must be one of the $11$ out of $50$ remaining cards with the same suit, etc. The reason the the numerator and denominator keep decrementing is because each time we choose a card, it takes one card away from the pool of the suit and the pool of total cards. So we should get the probability to be $$\prod_{n=1}^4 \frac{13-n}{52-n}$$ which equates to $\frac{33}{16660}$.

Answer 3

Let's use binomial coefficients to determine the number of ways that this is possible. Then I'll leave it to you to [use binomial coefficients to] divide this by the total number of outcomes to arrive at a probability.

For a 5-card poker hand, this is how we can draw n cards of the same suit.

$${13\choose n} \cdot{39 \choose 5-n} $$

The left factor is the chosen/given suit, and the right factor is all three of the other suits.

If we set $n=5$, that will give us the number of ways to draw all 5 in one of the suits. As there are 4 suits, we multiply the result by 4.

Answer 4

Hint -

Probability of getting 5 cards of same suit -

$$\frac{\binom {13}5}{\binom{52}{5}}$$

Now multiply with 4 as we have 4 suits.

$$4 \times \frac{\binom {13}5}{\binom{52}{5}}$$

Answer 5

Q: ,”What is the probability of drawing 5 cards from a deck of 52 that will have the same suit?”

There are 4 different suits in a deck of cards. Spades, clubs, hearts, and diamonds. There are 13 cards in each suit.

Let’s first find the probability of picking 5 cards of a specific suit in a row. Say we choose spades.

Drawing the 1st card: P(picking a spade) = ¹³⁄₅₂

Drawing the 2nd card: P(picking a spade) = ¹²⁄₅₁. Since we are assuming there is no replacement, the total number of spades decreases after each draw. (13 - 1 = 12) and the total amount of cards in the deck decreases thus the denominator decreases (52 - 1 = 51)

Drawing the 3rd card: P(picking a spade) = ¹¹⁄₅₀

Drawing the 4th card: P(picking a spade) = ¹⁰⁄₄₉

Drawing the 5th card: P(picking a spade) = ⁹⁄₄₈

Multiply all of these together as we need to find the probability of all of these occurring.

(¹³⁄₅₂ x ¹²⁄₅₁ x ¹¹⁄₅₀ x ¹⁰⁄₄₉ x ⁹⁄₄₈) = ³³⁄₆₆₆₄₀)

However, this is not our final answer.

³³⁄₆₆₆₄₀ is the probability of drawing 5 of a specific suit in a row, in this case I said spades.

However, the question is asking us what the probability of drawing 5 cards such that the cards can be a matching group of ANY suit.

This means your group could be 5 diamonds or 5 spades or 5 hearts or 5 clubs.

So therefore, we must add ³³⁄₆₆₆₄₀ four times in order to account for the fact that the matching group of five cards can be from any suit.

Final step: ³³⁄₆₆₆₄₀ + ³³⁄₆₆₆₄₀ + ³³⁄₆₆₆₄₀ + ³³⁄₆₆₆₄₀ = ³³⁄₁₆₆₆₀

Therefore, the probability of drawing 5 cards from a deck of 52 that will have the same suit = ³³⁄₁₆₆₆₀