Calculating the Probability of Drawing a Diamond, Club, and Queen from a Standard Deck of Cards

Given a standard deck of 52 cards, what is the likelihood of drawing a diamond, a club, and a queen in a specific sequence? In this article, we will break down the problem step-by-step and calculate the overall probability.

Step-by-Step Calculation

To find the probability of drawing a diamond, a club, and a queen in that specific order, we start by identifying the individual probabilities of each draw without replacement:

1. Probability of Drawing a Diamond First

There are 13 diamonds in a deck of 52 cards.

[ P(text{Diamond first}) frac{13}{52} frac{1}{4} ]

2. Probability of Drawing a Club Second

After drawing a diamond, there are now 51 cards left in the deck, and 13 clubs remain.

[ P(text{Club second | Diamond first}) frac{13}{51} ]

3. Probability of Drawing a Queen Third

After drawing a diamond and a club, there are 50 cards left in the deck, and 4 queens remain.

[ P(text{Queen third | Diamond first and Club second}) frac{4}{50} frac{2}{25} ]

Calculating the Overall Probability

Now, we can multiply these probabilities together to find the overall probability of drawing a diamond, a club, and a queen in that specific order:

[ P(text{Diamond, Club, and Queen}) P(text{Diamond first}) times P(text{Club second | Diamond first}) times P(text{Queen third | Diamond first and Club second}) ] [ P(text{Diamond, Club, and Queen}) frac{1}{4} times frac{13}{51} times frac{2}{25} ]

Substituting in the values and simplifying:

[ P(text{Diamond, Club, and Queen}) frac{1 times 13 times 2}{4 times 51 times 25} frac{26}{5100} ]

To further simplify:

[ P(text{Diamond, Club, and Queen}) frac{13}{2550} ]

Thus, the probability of drawing a diamond, a club, and a queen in that specific order is:

[ boxed{frac{13}{2550}} ]

Alternative Approach

Another way to solve this problem is by considering the number of favorable and exhaustive cases:

Favorable Cases

The number of favorable cases for drawing a diamond is 12 (excluding the Queen of Diamonds).

The number of favorable cases for drawing a club is 12 (excluding the Queen of Clubs).

The number of favorable cases for drawing a queen is 4.

Exhaustive Cases

The total number of ways to draw 3 cards out of 52 is given by the combination formula ( binom{52}{3} ):

[ binom{52}{3} frac{52 times 51 times 50}{1 times 2 times 3} 22100 ]

The probability of drawing one diamond, one club, and one queen:

[ P(text{Diamond, Club, and Queen}) frac{12 times 12 times 4}{22100} 0.026 ]

This confirms our previous calculation, showing the probability of drawing a diamond, a club, and a queen in that specific order is indeed ( frac{13}{2550} ).