Question

FAST PLEASE

A standard deck of 52 playing cards contains 13 cards in each of four suits: hearts, diamonds, clubs, and spades. Four cards are drawn from the deck at random. What is the approximate probability that exactly three of the cards are diamonds? 1% 4% 11% 44%

Answer 1

4% (four percent)

Answer 2

Answer: B) 4%

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Step-by-step explanation:

We'll involve the nCr combination formula here. This is because order doesn't matter.

There are n = 13 diamonds and we want to select exactly r = 3 of them.

So,

`_n C _r = (n!)/(r!*(n-r)!)\\\\_(13) C _(3) = (13!)/(3!*(13-3)!)\\\\_(13) C _(3) = (13!)/(3!*10!)\\\\_(13) C _(3) = (13*12*11*10!)/(3!*10!)\\\\_(13) C _(3) = (13*12*11)/(3!)\\\\_(13) C _(3) = (13*12*11)/(3*2*1)\\\\_(13) C _(3) = (1716)/(6)\\\\_(13) C _(3) = 286\\\\`

There are 286 ways to pick the three diamond cards. Then there are 52-13 = 39 ways to pick the fourth card that is either a spade, club of heart.

So we have 286*39 = 11,154 ways to pick the four cards given the conditions your teacher set.

This is out of 52C4 = 270,725 ways to pick four cards (use the nCr formula above with n = 52 and r = 4).

Divide the two values to get the answer:

(11,154)/(270,725) = 0.04120048019208

that rounds to 0.04 which converts to 4%

So there's roughly a 4% chance of getting exactly 3 diamond cards.