Question

out of a deck of 52 cards, what is the probability of drawing a club two times in a row without replacement?

Answer 1

To find the probability of drawing a club two times in a row without replacement from a standard deck of 52 cards, we'll follow these steps:

1. **Calculate the probability of drawing a club on the first draw:**
There are 13 clubs in a deck of 52 cards. Therefore, the probability of drawing a club on the first draw is:
\[
P(\text{first club}) = \frac{13}{52} = \frac{1}{4}
\]

2. **Calculate the probability of drawing a club on the second draw, given that the first draw was a club:**
If the first card drawn is a club, there are now 12 clubs left in a deck of 51 cards (since one card has been removed). Therefore, the probability of drawing a club on the second draw is:
\[
P(\text{second club | first club}) = \frac{12}{51}
\]

3. **Determine the overall probability of both events happening:**
To find the total probability of both events (drawing a club on the first draw and then again on the second draw), we multiply the probabilities of the two individual events:
\[
P(\text{two clubs in a row}) = P(\text{first club}) \times P(\text{second club | first club}) = \frac{13}{52} \times \frac{12}{51}
\]

Calculating this gives:
\[
P(\text{two clubs in a row}) = \frac{1}{4} \times \frac{12}{51} = \frac{12}{204} = \frac{1}{17}
\]

So, the probability of drawing a club two times in a row without replacement is \(\frac{1}{17}\).