Question

You are dealt five cards from a standard and shuffled deck of playing cards. Note that a standard deck has 52 cards and four of those are kings. What is the probability that you'll have at least two kings in your hand?

Answer 1

Answer: 0.041684 approximately

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Step-by-step explanation

The nCr combination formula will be used. That formula is

`_n C_r = (n!)/(r!*(n-r)!)`

If you plugged n = 52 and r = 5 into that formula, then you should get the result 2,598,960 which is a little under 2.6 million. This is the number of possible five-card hands. The order of the cards doesn't matter. I'll skip showing the scratch work and leave that for the student to do.

We'll return to this value later.

The phrasing "at least 2 kings" means "2 or more kings".

Let's consider three cases:

1. There are exactly 2 kings in the hand.
2. There are exactly 3 kings in the hand.
3. There are all 4 kings in the hand.

The next 3 sections will cover these cases.

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Case 1) Exactly 2 kings

There are 4 kings to pick from and 4C2 = 6 ways to pick a pair of kings (again order doesn't matter). The 4C2 refers to the combination formula when the inputs are n = 4 and r = 2.

After the 2 kings are chosen, there are 48 cards that aren't a king (since 52-4 = 48) and 3 slots for them. That gives 48C3 = 17296 ways to pick the non-king cards.

Overall we have 6*17296 = 103776 different hands when exactly 2 kings are selected.

This value will be used later. Be sure to circle it or highlight it in some way.

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Case 2) Exactly 3 kings

We have 4C3 = 4 ways to pick 3 kings. This is equivalent to 4 ways to leave a particular king out.

Then we have 48C2 = 1128 ways to pick the last two cards that aren't a king.

We have 4*1128 = 4512 different hands for case 2.

This value will be used later.

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Case 3) All 4 kings

There's only one way to select all four kings. Note how 4C4 = 1.

Then there are 48 choices for the fifth slot.

We have 1*48 = 48 different hands possible when all 4 kings are selected.

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The results of the previous 3 sections were:

103776

4512

48

Add them up: 103776+4512+48 = 108336

This is the number of hands that have at least two kings.

Divide this over the 52C5 = 2,598,960 value we found earlier (recall this is the number of five-card hands).

108336/2598960 = 0.041684 approximately

There's about a 4.17% chance of getting at least two kings.

Normally I would go with the fraction form for the probability; however, the values are so large that it might not seem that intuitive. So instead it might be best to go with the approximate decimal form. Be sure to ask your teacher which format she or he prefers. Round that approximate value however instructed.