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?

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

we have three similar triangles, because each has a right angle and shares an angle.   let's write the angles in order: opposite to short leg, long leg, hypotenuse.

cab similar to bad similar to cbd

or as ratios,

ca: ab: cb = ba: ad: bd = cb: bd: cd

we also know

ac = ad + cd

(ad+cd): ab: cb = ba: ad: bd = cb: bd: cd

(ad+cd)/cb=cb/cd

we have cb=6, ad=5 and seek x=cd.

we reject the negative root and conclude x=4

answer: 4

c

step-by-step explanation: