# Two cards are chosen at random from a deck of 52 cards. What is the probability of getting a diamond, replacing the card, then getting a red jack?

Dec 18, 2016

The probability of getting a diamond, replacing the card, then getting a red jack is $\frac{1}{104}$.

#### Explanation:

As there are $13$ diamond cards in a deck of $52$ cards,

probability of getting a diamond is $\frac{13}{52} = \frac{1}{4}$

As we replace the card, the deck is as it is and

in deck of $52$ cards there are $2$ red jacks - one of diamond and another of hearts

Hence, probability of getting a red jack is $\frac{2}{52} = \frac{1}{26}$

As the two events are independent (due to replacement of card)

$P \left(A \cup B\right) = P \left(A\right) \times P \left(B\right) = \frac{1}{4} \times \frac{1}{26} = \frac{1}{104}$

Hence, the probability of getting a diamond, replacing the card, then getting a red jack is $\frac{1}{104}$.