If three cards are drawn one by one without replacement from a well shuffled pack of 52 cards. What is the probability that the first card is jack, second is queen and third is aga... If three cards are drawn one by one without replacement from a well shuffled pack of 52 cards. What is the probability that the first card is jack, second is queen and third is again jack?
Understand the Problem
The question is asking for the probability of drawing three specific cards from a deck: the first being a Jack, the second a Queen, and the third again a Jack. This requires understanding of combinatorial probabilities.
Answer
The probability is $\frac{2}{55125}$.
Answer for screen readers
The total probability is:
$$ P(total) = \frac{4}{52} \times \frac{4}{51} \times \frac{3}{50} = \frac{48}{132600} = \frac{2}{55125} $$
Steps to Solve
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Calculate the probability of drawing the first Jack
The probability of drawing a Jack from a deck of 52 cards is given by the formula: $$ P(Jack_1) = \frac{4}{52} $$
There are 4 Jacks in a 52-card deck. -
Calculate the probability of drawing the Queen next
After drawing a Jack, there are now 51 cards left. The probability of then drawing a Queen is: $$ P(Queen) = \frac{4}{51} $$
Again, there are still 4 Queens in the remaining deck. -
Calculate the probability of drawing the second Jack
Now, after drawing a Jack and a Queen, there are 50 cards remaining in the deck. The probability of drawing a Jack again is: $$ P(Jack_2) = \frac{3}{50} $$
There are now 3 Jacks left since one was already drawn. -
Calculate the total probability
To find the overall probability of these events happening in sequence, multiply the individual probabilities together: $$ P(total) = P(Jack_1) \times P(Queen) \times P(Jack_2) $$
The total probability is:
$$ P(total) = \frac{4}{52} \times \frac{4}{51} \times \frac{3}{50} = \frac{48}{132600} = \frac{2}{55125} $$
More Information
This probability demonstrates how drawing specific cards in a defined sequence affects the likelihood due to reduced card counts in the deck. Such calculations are fundamental in probability theory and combinatorial analysis.
Tips
- Not adjusting the total number of cards in the deck after each draw.
- Confusing the order of card draws, which can lead to incorrect probabilities.
- Forgetting to multiply probabilities for sequential events.
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