Card Probability: Calculating the Odds of Drawing Specific Cards in Sequence

Let's dive into a fascinating problem in probability: calculating the likelihood of choosing a six, a seven, an eight, and a nine in that exact order from a standard deck of 52 cards, without replacement. This problem highlights the intricacies of probability and how the sequence of events affects the outcome.

Understanding the Basics of Probability

In probability theory, the probability of an event occurring is defined as the number of favorable outcomes divided by the total number of possible outcomes. For a standard deck of 52 cards, the total number of possible outcomes is 52 for the first draw, 51 for the second, 50 for the third, and 49 for the fourth. This is because the drawing is done without replacement, meaning each card drawn is removed from the deck and not returned.

Breaking Down the Calculation

Step 1: Probability of Drawing a Six First

When drawing a card from a full deck, the number of sixes is 4 (one six for each suit: hearts, diamonds, clubs, spades). Therefore, the probability of drawing a six first is:

1. Probability of Drawing a Six 4/52

Step 2: Probability of Drawing a Seven Next

Now, one card (a six) has been removed from the deck, so there are 51 cards remaining. The number of sevens is still 4. Hence, the probability of drawing a seven next is:

2. Probability of Drawing a Seven 4/51

Step 3: Probability of Drawing an Eight Subsequently

Two cards have now been removed (a six and a seven), leaving 50 cards in the deck. The number of eights is again 4. The probability of drawing an eight next is:

3. Probability of Drawing an Eight 4/50

Step 4: Probability of Drawing a Nine Fourth

Three cards have been removed (a six, a seven, and an eight), so there are 49 cards left. The number of nines remains 4. The probability of drawing a nine last is:

4. Probability of Drawing a Nine 4/49

Combining Probabilities

Since these events are dependent on each other (due to the lack of replacement), we multiply the probabilities of each step to find the overall probability of drawing a six, a seven, an eight, and a nine in that specific order:

Probability (4/52) * (4/51) * (4/50) * (4/49)

The Final Calculation

Let's calculate this step-by-step:

First, we simplify the fractions: 1. 4/52 1/13 2. 4/51 4/51 3. 4/50 2/25 4. 4/49 4/49

Now, multiplying these probabilities together:

Probability (1/13) * (4/51) * (2/25) * (4/49)

Breaking it down further:

Probability 1 * 4 * 2 * 4 / (13 * 51 * 25 * 49)

Probability 32 / (16575 * 13)

Probability 32 / 215500

Finally, simplifying the fraction:

Probability ≈ 0.000149 or approximately 0.0149%

This result shows the rarity of the specific sequence occurring, demonstrating the complexity of drawing cards in such a precise order without replacement.

Conclusion

This problem in probability not only showcases the importance of understanding dependent events but also highlights the power of sequential and conditional probability. It is a great example for illustrating how the number of possible outcomes diminishes with each draw, significantly affecting the overall probability of specific events.