Question

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?

Answer 1

`\bf\textbf{The required Probability = }(1)/(8)`

Step-by-step explanation :

Number of heart cards = 13

Number of club cards = 13

Number of diamond cards = 1 3

Number of spades card = 13

`\text{Number of ways of drawing 1 heart card from 13 cards of heart = }_(1)^(13)\textrm{C}=13\\\\\text{Also,no of ways 1 card can be drawn from a pack of 52 cards = }_(1)^(52)\textrm{C}=52\\\\\text{The probability of drawing a heart card from a pack of 52 cards = }(13)/(52)=(1)/(4)`

There are 13 spades card and 13 club cards

⇒ Number of black cards = 13 + 13

= 26

`\text{Number of ways of drawing 1 black card from 26 black cards = }_(1)^(26)\textrm{C}=26\\\\\text{Also,no of ways 1 card can be drawn from a pack of 52 cards = }_(1)^(52)\textrm{C}=52\\\\\text{The probability of drawing a black card from a 26 black cards = }(26)/(52)=(1)/(2)`

`\textbf{Now, The probability that a black card is chosen first}\\\textbf{and a heart is chosen second = }(1)/(4)* (1)/(2)\\\\\bf\implies\textbf{The required Probability = }(1)/(8)`

Answer 2

There are 13 heart cards

Now 1 heart card can be drawn from a pack of 13 heart cards in 13C1 ways

13C1=13

Also,1 card can be drawn from a pack of 52 cards in 52C1 ways

52C1=52

The probability of drawing a heart card from a pack of 52 cards is

13/52=1/4

there are 26 black cards in total

(13 spades +13 clubs)

Now 1 black card can be drawn from a pack of 26 black cards in 26C1 ways

26C1=26

Also,1 card can be drawn from a pack of 52 cards in 52C1 ways

52C1=52

The probability of drawing a black card from a pack of 52 cards is

26/52=1/2

Now as both the task are performed one after the other and after replacement so the probability of drawing a heart after a black card is

=1/2 * 1/4

=1/8