Probability of Drawing Three Clubs in Succession from a Deck of Cards: A Detailed Calculation

The game of cards is not only thrilling but also a rich reservoir for probability problems. One such classic problem is determining the likelihood of drawing three clubs in succession without replacement from a standard deck of 52 playing cards.

Understanding the Problem

The problem at hand is to find the probability that the first three cards drawn from a standard deck are all clubs. A standard deck contains 52 cards, with 13 of these being clubs. The crucial aspect here is the absence of replacement, meaning each card drawn affects the subsequent probability.

Step-by-Step Calculation

Step 1: Count the Total Number of Clubs

A standard deck has 13 clubs. Let's denote this as Cl.

Step 2: Calculate the Probability for Each Draw

1. First Card Drawn: The probability of drawing a club as the first card is calculated as:

P(1st card is a club) 13/52 1/4

2. Second Card Drawn: Assuming the first card drawn was a club, there are now 12 clubs left and 51 total cards remaining. The probability of drawing another club is:

P(2nd card is a club | 1st card is a club) 12/51

3. Third Card Drawn: If the first two cards drawn were clubs, then there are now 11 clubs left and 50 total cards remaining. The probability of drawing a club as the third card is:

P(3rd card is a club | 1st and 2nd cards are clubs) 11/50

Step 3: Multiply the Probabilities

The overall probability that the first three cards drawn are clubs is the product of the individual probabilities:

P(first three cards are clubs) P(1st card is a club) * P(2nd card is a club | 1st card is a club) * P(3rd card is a club | 1st and 2nd cards are clubs)

(1/4) * (12/51) * (11/50)

Step 4: Simplify the Expression

First, calculate the numerator:

Numerator 13 * 12 * 11 1560

Then, calculate the denominator:

Denominator 52 * 51 * 50 132600

Thus, the overall probability is:

P(first three cards are clubs) 1560/132600

Step 5: Reduce the Fraction

To simplify the fraction, we find the greatest common divisor (GCD). Both 1560 and 132600 are divisible by 12:

(1560/12) / (132600/12) 130/11050

Hence, the probability that the first three cards drawn are clubs is:

boxed{130/11050}

Conclusion

From this detailed calculation, we can see how the probability of drawing three clubs in succession decreases with each draw due to the lack of replacement. Understanding such probability problems can provide valuable insights into the workings of discrete probability and combinatorics.

Related Concepts and Further Reading

1. Probability: A branch of mathematics dealing with the likelihood of events occurring. Further reading: Probability on Wikipedia

2. Deck of Cards: Understanding the composition and distribution of cards in a standard deck. Further reading: Deck of Cards on MathIsFun

3. Drawing Clubs: Exploring the probabilities associated with specific card draws. Further reading: Card Draw Probabilities on MathCounterpart