Three cards are dealt at random from a standard deck of 52 cards.  What is the probability that the first card is a Jack, the second card is a Queen, and the third card is a King? Assuming the dealing is done without replacement.
The probability is $\dfrac{4}{52} \times \dfrac{4}{51} \times \dfrac{4}{50} = \boxed{\frac{8}{16575}}$.
\frac{8}{16575}
Level 3
Counting & Probability
Problem: Three cards are dealt at random from a standard deck of 52 cards.  What is the probability that the first card is a Jack, the second card is a Queen, and the third card is a King? Assuming the dealing is done without replacement.
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the probability of drawing a Jack first, a Queen second, and a King third from a standard deck of 52 cards without replacement, we need to calculate the probability of each event occurring in sequence.

First, consider the total number of cards in the deck, which is 52. The probability of drawing a Jack first is the number of Jacks in the deck divided by the total number of cards. There are 4 Jacks in a standard deck, so the probability of drawing a Jack first is:

P(Jack first) = 4/52.

After drawing a Jack, there are now 51 cards left in the deck. The probability of drawing a Queen second is the number of Queens in the deck divided by the remaining number of cards. There are still 4 Queens in the deck, so the probability of drawing a Queen second is:

P(Queen second | Jack first) = 4/51.

Next, after drawing a Jack and a Queen, there are now 50 cards left in the deck. The probability of drawing a King third is the number of Kings in the deck divided by the remaining number of cards. There are still 4 Kings in the deck, so the probability of drawing a King third is:

P(King third | Jack first, Queen second) = 4/50.

To find the overall probability of all three events occurring in sequence, we multiply the probabilities of each individual event:

P(Jack first, Queen second, King third) = P(Jack first) * P(Queen second | Jack first) * P(King third | Jack first, Queen second).

Substituting the probabilities we calculated:

P(Jack first, Queen second, King third) = (4/52) * (4/51) * (4/50).

Calculating this gives:

= (4 * 4 * 4) / (52 * 51 * 50) = 64 / 132600.

To simplify this fraction, we divide both the numerator and the denominator by 16:

= 4 / 8287.5.

Since we need the final answer in a specific form, we convert this to a fraction with integer values. The final answer is expressed as:

= 8 / 16575.<|END_OF_LATENT|> The final answer is $\frac{8}{16575}$.