If 4 cards are drawn at random from a standard deck of cards, what are the odds, or probability, that the first three will be of the same suit and the fourth of another suit?

Question

If 4 cards are drawn at random from a standard deck of cards, what are the odds, or probability, that the first three will be of the same suit and the fourth of another suit?

Answer 1

First card: any card (52/52)

Second card: one of 12 remaining cards of the same suit (12/51)
third card: one of 11 remaining cards of the same suit (11/50)
fourth card: one of 39 cards of any other suit (39/49)

Since all these are required events, use the multiplication rule to get the combined probability of all four events happening.

Odds are P(E)/(1-P(E)) where P(E) is the probability of an event taking place.

Answer 2

To find the probability of drawing the first three cards of the same suit and the fourth card of another suit, we need to calculate the number of favorable outcomes and the total number of possible outcomes.

Let's break down the problem step by step:

Step 1: Calculate the number of favorable outcomes.
To have the first three cards of the same suit (let's say hearts), we need to select 3 out of the 13 hearts cards in the deck. Therefore, the number of ways to choose these three cards is given by the combination formula: C(13, 3) = 286.

Now, for the fourth card to be of a different suit (let's say diamonds, which is a different suit from hearts), we need to select 1 card out of the 13 diamonds cards in the deck. Thus, the number of ways to choose this single card is C(13, 1) = 13.

Since we need these two events (getting three hearts and one diamond) to occur simultaneously, we can multiply the two values together: 286 * 13 = 3718.

Therefore, the number of favorable outcomes is 3718.

Step 2: Calculate the total number of possible outcomes.
When drawing 4 cards from a standard deck of 52 cards, without replacement, the total number of possible outcomes can be obtained by calculating the combination of the 52 cards taken 4 at a time: C(52, 4) = 270,725.

Step 3: Calculate the probability.
The probability is given by the ratio of the number of favorable outcomes to the total number of possible outcomes:

Probability = Number of favorable outcomes / Total number of possible outcomes

Plugging in the values, we have:

Probability = 3718 / 270725 ≈ 0.0137 ≈ 1.37%

Therefore, the probability that the first three cards will be of the same suit and the fourth card of another suit is approximately 1.37%.