Probability of Drawing the King of Diamonds from a 52-Card Deck

The problem of determining the probability of drawing the King of Diamonds from a 52-card deck after initially drawing one card without replacement is a classic example that explores fundamental concepts in probability theory. This article will break down the process step-by-step, explaining the reasoning behind each calculation.

Initial Situation

A standard 52-card deck consists of 52 cards, including 1 King of Diamonds. The first draw can result in either one of two scenarios:

You draw the King of Diamonds. You do not draw the King of Diamonds.

Calculating Probabilities

Let's calculate the probability for each scenario:

Scenario A (First card not the King of Diamonds): Probability of this happening: 1 - (1/52) 51/52 If the first card is not the King of Diamonds, there are now 51 cards left, including the King of Diamonds. The probability of drawing the King of Diamonds in this case is 1/51. Therefore, the combined probability for this scenario is: (51/52) * (1/51) 1/52 Scenario B (First card is the King of Diamonds): Probability of this happening: 1/52 If the first card is the King of Diamonds, there are now 51 cards left, excluding the King of Diamonds. The probability of drawing the King of Diamonds in this case is 0. Therefore, the combined probability for this scenario is: (1/52) * 0 0

Using the law of total probability, we can now sum the probabilities of both scenarios:

P(King of Diamonds on 2nd draw) (51/52) * (1/51) (1/52) * 0 1/52

Thus, the probability of drawing the King of Diamonds on the second draw is 1/52.

Additional Example: Drawing Two Diamonds in a Row

Let's explore another scenario: what is the probability of drawing two diamonds in a row from a 52-card deck?

First Draw

There are 13 diamonds in a 52-card deck, so the probability of drawing a diamond on the first draw is:

13/52 1/4

Second Draw

After drawing one diamond, there are 12 diamonds left and 51 total cards. The probability of drawing a diamond on the second draw is:

12/51 4/17

Including Both Draws

The probability of drawing two diamonds in a row is the product of the individual probabilities:

(1/4) * (4/17) 1/17 ≈ 5.9%

Therefore, the probability of drawing two diamonds in a row is approximately 5.9%.