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} ).