in a standard deck of cards there are 13 spades, 13 clubs, 13 hearts, and 13 diamonds. The spades and the clubs are black and the hearts and the diamonds are red. If two cards are chosen at random from a deck, one at a time, and replaced after each pick, what is the probability that a black card is chosen first and a heart is chosen second?
1/8, 1/2, 2/3, 3/4

Black card is 1/2 probability. After that, the probability of choosing a heart is 1/4.

Therefore, if you multiply these numbers, you get the answer of 1/8.

Answer Link

ColinJacobus ColinJacobus · Answer 2 · 2019-05-29T17:35:33+02:00

Answer: The correct option is (A) [tex]\dfrac{1}{8}.[/tex]

Step-by-step explanation: Given that in a standard deck of cards there are 13 spades, 13 clubs, 13 hearts, and 13 diamonds. The spades and the clubs are black and the hearts and the diamonds are red.

We are to find the probability that a black card is chosen first and a heart is chosen second, if two cards are chosen at random from a deck, one at a time, and replaced after each pick.

Let S be the sample space for the experiment of choosing a card from the deck of 52 cards.

Then, n(S) = 52.

Let A be the event of choosing a black card and B be the event of choosing a heart card.

So, n(A) = 13 + 13 = 26, n(B) = 13.

Therefore, we get

[tex]P(A)=\dfrac{n(A)}{n(S)}=\dfrac{26}{52}=\dfrac{1}{2},\\\\\\P(B)=\dfrac{n(B)}{n(S)}=\dfrac{13}{52}=\dfrac{1}{4}.[/tex]

Since A and B are independent events, so the required probability that a black card is chosen first and a heart is chosen second is given by

[tex]P(A\cap B)\\\\=P(A)\times P(B)\\\\=\dfrac{1}{2}\times \dfrac{1}{4}\\\\\\=\dfrac{1}{8}.[/tex]

Thus, option (A) is CORRECT.