# What are the odds of predicting four cards dealt from a deck of twelve

**URL:** <https://boards.straightdope.com/t/what-are-the-odds-of-predicting-four-cards-dealt-from-a-deck-of-twelve/791360>\
**Category:** Factual Questions\
**Created:** [July 16, 2017, 12:53am UTC](https://boards.straightdope.com/t/what-are-the-odds-of-predicting-four-cards-dealt-from-a-deck-of-twelve/791360 "2017-07-16T00:53:54Z")\
**Posts on this page:** 8\
**Page:** 2

<div class="post-metadata">

**Author:** ![Whack-a-Mole](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/whack-a-mole/32/141_2.png) [@Whack-a-Mole](https://boards.straightdope.com/u/Whack-a-Mole)\
**Post date:** [July 16, 2017, 9:21pm UTC](https://boards.straightdope.com/t/what-are-the-odds-of-predicting-four-cards-dealt-from-a-deck-of-twelve/791360/21 "2017-07-16T21:21:21Z")

</div>

Question: If we have cards numbered 1-9 how do the odds change when we have four each of cards 1-9? (Ace = 1)

After that we need to figure in the face cards but one step at a time.

---

<div class="post-metadata">

**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\
**Post date:** [July 17, 2017, 1:21am UTC](https://boards.straightdope.com/t/what-are-the-odds-of-predicting-four-cards-dealt-from-a-deck-of-twelve/791360/22 "2017-07-17T01:21:42Z")

</div>

Are there 10s in the deck?

---

<div class="post-metadata">

**Author:** ![md2000](https://avatars.discourse-cdn.com/v4/letter/m/73ab20/32.png) [@md2000](https://boards.straightdope.com/u/md2000)\
**Post date:** [July 17, 2017, 5:31am UTC](https://boards.straightdope.com/t/what-are-the-odds-of-predicting-four-cards-dealt-from-a-deck-of-twelve/791360/23 "2017-07-17T05:31:30Z")

</div>

Ok - pick 4 cards. 40 cards (A-9) plus 3 face cards. Suite does not matter. Order does matter.  
For no face cards - 40x39x38x37 =2,024,640 hands.

For 1 face card - 40x39x38 = 59,280 \<== Three non-face cards.  
Now pick one of 3 face cards; then that could be first, second, third or fourth. (FNNN, NFNN, NNFN, NNNF)  
59,280x3x4 = 711,360

For two face cards - 40x39 = 1560 for the two number cards.  
Pick 2 of 3 face cards (i.e. don’t pick 1 so 3 choices) You can pick A then B or B then A.  
Hands: ABNN, ANBN, ANNB, NABN, NANB, NNAB; and swap A,B -BANN, BNAN, etc. Total of 12.  
1560x3x12 = 56,160

Total possible hand choices - sum of the above - 2,736,000  
So odds of picking the right hand - 1:2736,000

Then comes the fun part - eliminate hands where suite doesn’t matter … good grief…  
(I.e. a had with 2C 2H is same as 2H 2C)

I’m thinking its something along the lines of -  
Number of hands where two number are the same divided by 2, and 2 pairs divided by 16, number of hands where 3 numbers same divided by 6, and there are 10 hands of 4 that can be picked 160 different ways…

Not a trivial problem.

---

<div class="post-metadata">

**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\
**Post date:** [July 17, 2017, 6:02am UTC](https://boards.straightdope.com/t/what-are-the-odds-of-predicting-four-cards-dealt-from-a-deck-of-twelve/791360/24 "2017-07-17T06:02:13Z")

</div>

Still not sure what cards are in the deck, but for the sake of argument let’s use the 43-card deck described in Post #8.

In that case, nothing is substantially different from the previous calculation. There are 43×42×41×40 ways to deal 4 cards. A, A, A, A corresponds to 24 of them, so you would not want to guess that. A, 2, 3, 4 has a 1-in-11570 chance of winning.

ETA the worst guess is something like A, J, Q, K, of course.

---

<div class="post-metadata">

**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\
**Post date:** [July 17, 2017, 6:19am UTC](https://boards.straightdope.com/t/what-are-the-odds-of-predicting-four-cards-dealt-from-a-deck-of-twelve/791360/25 "2017-07-17T06:19:43Z")

</div>

**md2000** , you are just making things unnecessarily complicated 🙂

And missing a bunch of permutations; there are nearly 3 million ways of drawing 4 cards, not 2,7

---

<div class="post-metadata">

**Author:** ![Whack-a-Mole](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/whack-a-mole/32/141_2.png) [@Whack-a-Mole](https://boards.straightdope.com/u/Whack-a-Mole)\
**Post date:** [July 17, 2017, 6:31am UTC](https://boards.straightdope.com/t/what-are-the-odds-of-predicting-four-cards-dealt-from-a-deck-of-twelve/791360/26 "2017-07-17T06:31:41Z")

</div>

> [@DPRK](#):
>
> Are there 10s in the deck?

Actually there are no tens (since he wants the guess to be a four digit guess) but I felt bad changing the parameters again. I figured the math would be the same and easy to recalculate without the tens if we ever figured the math out here.

To be clear the cards are:

- 1-9, four of each number (Aces are “1”)

- Jack, Queen, King (one each)

You can see him do it here: [https://youtu.be/xdsakEafFRU?t=46](https://youtu.be/xdsakEafFRU?t=46)

---

<div class="post-metadata">

**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\
**Post date:** [July 17, 2017, 3:43pm UTC](https://boards.straightdope.com/t/what-are-the-odds-of-predicting-four-cards-dealt-from-a-deck-of-twelve/791360/27 "2017-07-17T15:43:00Z")

</div>

Well, with only 39 cards, there are 1974024 different results, so A,2,3,4 will match approximately 1 time in 7711.

Hijack: obviously A = 1, but who started to print the letter A on the card? I have a French-style Tarot deck (78 cards) where it just says “1”.

Correction/clarification: way above I wrote, “There is a 64% chance…” Just to be clear, that is the chance of an arbitrary hand containing no pair or above; that was for the 43-card deck though.

---

<div class="post-metadata">

**Author:** ![Trinopus](https://avatars.discourse-cdn.com/v4/letter/t/2bfe46/32.png) [@Trinopus](https://boards.straightdope.com/u/Trinopus)\
**Post date:** [July 18, 2017, 12:02am UTC](https://boards.straightdope.com/t/what-are-the-odds-of-predicting-four-cards-dealt-from-a-deck-of-twelve/791360/28 "2017-07-18T00:02:14Z")

</div>

> [@DPRK](#):
>
> . . . There are 43×42×41×40 ways to deal 4 cards. A, A, A, A corresponds to 24 of them . . .

Aha! The little light bulb comes on. I don’t want to _divide_ by 24; I want to _subtract_ 23! (And also for all possible combinations like AAA2, AA23, AA22, and so on.) That’s how to get the total number of distinct possible deals.

[Previous page](https://boards.straightdope.com/t/what-are-the-odds-of-predicting-four-cards-dealt-from-a-deck-of-twelve/791360.md?page=1)