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QuestionSuppose that 6 cards are drawn from a well-shuffled deck of 52 cards. What is the probability that all 6 are not face cards?

Studdy Solution

STEP 1

What is this asking? What are the chances of pulling 6 cards from a deck and *none* of them being a Jack, Queen, or King? Watch out! Don't forget there are 4 suits, so there are a total of 12 face cards!

STEP 2

1. Calculate the number of non-face cards.
2. Calculate the total number of ways to choose 6 cards.
3. Calculate the number of ways to choose 6 non-face cards.
4. Calculate the probability.

STEP 3

Alright, let's **start** by figuring out how many cards *aren't* face cards!
We know there are 52 total cards in a standard deck.
There are 3 face cards (Jack, Queen, King) in each of the 4 suits (hearts, diamonds, clubs, spades).

STEP 4

So, that's 3⋅4=123 \cdot 4 = 12 face cards.
Since there are 52 total cards, the number of non-face cards is 52−12=4052 - 12 = 40.
That's **40** non-face cards!

STEP 5

Now, imagine you're picking any 6 cards from the deck.
How many different ways can you do that?
This is a combination problem because the order doesn't matter.
The formula for combinations is (nk)=n!k!(n−k)!\binom{n}{k} = \frac{n!}{k!(n-k)!}, where nn is the total number of items and kk is the number you're choosing.

STEP 6

In our case, n=52n = 52 (total cards) and k=6k = 6 (cards we're picking).
So, the total number of ways to choose 6 cards is: (526)=52!6!(52−6)!=52!6!46!=52⋅51⋅50⋅49⋅48⋅476⋅5⋅4⋅3⋅2⋅1=20,358,520 \binom{52}{6} = \frac{52!}{6!(52-6)!} = \frac{52!}{6!46!} = \frac{52 \cdot 51 \cdot 50 \cdot 49 \cdot 48 \cdot 47}{6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1} = 20,358,520 That's a whopping **20,358,520** different ways!

STEP 7

Now, let's imagine we *only* want to pick from the non-face cards.
We already figured out there are 40 non-face cards.
We still want to pick 6.
So, using the same combination formula, with n=40n = 40 and k=6k = 6, we get: (406)=40!6!(40−6)!=40!6!34!=40⋅39⋅38⋅37⋅36⋅356⋅5⋅4⋅3⋅2⋅1=3,838,380 \binom{40}{6} = \frac{40!}{6!(40-6)!} = \frac{40!}{6!34!} = \frac{40 \cdot 39 \cdot 38 \cdot 37 \cdot 36 \cdot 35}{6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1} = 3,838,380 There are **3,838,380** ways to pick 6 non-face cards.

STEP 8

Finally, the probability is the number of ways to choose 6 non-face cards divided by the total number of ways to choose any 6 cards.

STEP 9

So, our probability is: 3,838,38020,358,520≈0.1885 \frac{3,838,380}{20,358,520} \approx 0.1885 That means there's roughly an **18.85%** chance of getting 6 non-face cards!

STEP 10

The probability that all 6 cards drawn are not face cards is approximately 0.1885 or 18.85%.

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