Point values in the game “Target”

Contents

1 Introduction
2 Setup
3 Graph
4 Results
5 Derivation
 5.1 Flushes
 5.2 Straights
 5.3 Straight Flushes
 5.4 Poker Hands
 5.5 Miscellaneous
6 About this project

1 Introduction

Target is a game where you draw cards to try to satisfy the target cards. Each of the cards has a separate point value, and I wondered how ”true” those points were. So, my goal was to calculate the number of 6 card hands (in the game your hand limit is 5, but you draw a card and then can turn sets in) that satisfy each of the conditions and see if there was a way to derive how points were allocated. It was also a good excuse to calculate (using combinatorics) the exact number of hands that satisfy each condition, and to check those numbers using a Python script.

2 Setup

Target cards are 0-9 in four different suits (purple, green, red, blue) plus 1 wild suit in each number. 0 and 9 have one of each suit and all other numbers have two of each suit for a total of 2(4 + 1) + 8(2 4 + 1) = 82 cards. The total number of possible hands is (82)
 6 = 350161812  .

3 Graph

The x-axis represents the probability that a random 6-card hand will satisfy the target card. The y-axis is just the number of cards you have to turn in to fulfill the target card, and the color of the dot (and the corresponding text) is how many points it’s worth. So, dots in the upper-left hand corner are the hardest to get and take the most cards, and you would expect these to be worth the most points (which they generally are).
pict
There are some interesting results here - all the 5 point targets are at the left and towards the top, as expected, but there are some interesting outliers. At 2 points, 5 odd cards seems undervalued; it’s the only target that takes more than 3 cards that’s only worth 2 points (and it has a pretty low probability to boot). 4 card straight including a 3 and skip straight both seem undervalued since mixed straight is more likely and is worth more points. 3 card straight including a 2 looks similarly undervalued. (or maybe 3 cards totalling = 4 is overvalued?)

One possibility is that the designers of the game looked at the probabilities with 5 card hands instead of 6 cards as I’ve done. Another is that they chose to fiddle with the point values to account for the fact that certain target cards are easier to turn in together - if you have 5 odd cards you have a pretty good chance of having a skip straight, etc.

4 Results

I was able to calculate almost all the results symbolically. Here are the results. Click on a hand type to see the derivation of that number. These results were derived by hand (except the ones in bold) and checked with a Python script.

Hand type PointsPossible handsProbability
Flush
3 card, specific suit 2 116241489 33.197%
4 card, any suit 3 122384088 34.951%
5 card, any suit 5 22272138 6.361%
Straight
3 card including a 2 2 72355734 20.664%
3 card including a 4 2 81222993 23.196%
3 card including a 5 2 81222993 23.196%
3 card including a 7 2 72355734 20.664%
4 card including a 3 3 36204327 10.339%
4 card including a 6 3 36204327 10.339%
5 card 5 15149349 4.326%
Straight Flush
3 card, specific suit 4 12588978 3.595%
3 card, any suit 3 45888079 13.105%
4 card, any suit 5 5054955 1.444%
Poker
2 pair 4 92445176 26.401%
3 of a kind 4 45800298 13.080%
Full house 5 12646280 3.612%
Misc
5 odd cards 2 35221706 10.059%
5 even cards 2 35221706 10.059%
3 cards totalling 4 3 73617980 21.024%
3 cards totalling 233 73617980 21.024%
Skip straight 3 35894502 10.251%
Mixed straight 4 62519949 17.855%

5 Derivation

5.1 Flushes

5.1.1 3 card flush, specific suit

Without loss of generality, let the suit be G(reen). There are 28 cards that can be G - 18 G and 10 W(ild), and so there are 82 - 28 = 54 cards that cannot be G. To avoid double-counting, we split up the possibilities:

So the total is 376740 + 5307120 + 29299725 + 81257904 = 116241489.
Back to table

5.1.2 4 card flush, any suit

We break it down by the number of wilds in the hand. Note that there are 10 wilds and 18 cards of each suit, which means there are 54 cards not in a given suit.

So, the total is 19440384+53660160+41571630+7156800+536760+18144+210 = 122384088.
Back to table

5.1.3 5 card flush, any suit

Similarly, we break it down by the number of wilds in the hand.

So, the total is 1924944+6952320+8482320+4357440+536760+18144+210 = 22272138.
Back to table

5.2 Straights

5.2.1 3 card including a 2

We break it down by the length of the straight:

So, the total is 1358127 + 6482268 + 19235394 + 45279945 = 72355734.
Back to table

5.2.2 3 card including a 4

We break it down by the length of the straight:

So, the total is 2184813 + 11324286 + 26370846 + 41343048 = 81222993.
Back to table

5.2.3 3 card including a 5

By symmetry this is the same number as 3 card straight including a 4, which is 81222993.
Back to table

5.2.4 3 card including a 7

By symmetry this is the same number as 3 card straight including a 2, which is 72355734.
Back to table

5.2.5 4 card including a 3

We break it down into the length of the straight:

So, the total is 1889568 + 8785179 + 5452920 + 7488288 + 2 6294186 = 36204327.
Back to table

5.2.6 4 card including a 6

By symmetry this is the same number as 4 card straight including a 3, which is 36204327.
Back to table

5.2.7 5 card

We break it down into the length of the straight:

So, the total is 2184813 + 12964536 = 15149349.
Back to table

5.3 Straight Flushes

5.3.1 3 card, specific suit

Since we’re dealing in a specific suit (let’s say G(reen)), there are 3 cards of that suit (including 1 wild) for each number except 0 and 9, and there are 2 cards for those. We break it down by the length of the straight flush:

So, the total is 3159 + 87156 + 1206468 + 11292195 = 12588978.
Back to table

5.3.2 3 card, any suit

You can perhaps do some sort of inclusion-exclusion argument where your sets are the hands that have 3 cards in any particular suit, but it gets very complicated very fast - open to suggestions.
Back to table

5.3.3 4 card, any suit

You can perhaps do some sort of inclusion-exclusion argument where your sets are the hands that have 4 cards in any particular suit, but it gets very complicated very fast - open to suggestions.
Back to table

5.4 Poker Hands

5.4.1 2 pair

We can make two pair by all the possibilities under Full House, plus the following:

So, the total is 12646280 + 3367296 + 76431600 = 92445176.
Back to table

5.4.2 3 of a kind

We break it down by the number of cards of the same number:

So, the total is 672 + 73738 + 2678284 + 211108 + 42836496 = 45800298.
Back to table

5.4.3 Full house

We break it down by the distribution of cards in the full house:

So, the total is 277156 + 211108 + 12158016 = 12646280.
Back to table

5.5 Miscellaneous

5.5.1 5 odd cards

We break it down by the number of odd cards:

This gives a total of 4496388 + 30725318 = 35221706.
Back to table

5.5.2 5 even cards

Since there are 41 even cards and 41 odd cards, by symmetry this is the same number as 5 odd cards, which is 35221706.
Back to table

5.5.3 3 cards totalling 4

We break it down by the lowest two cards in the hand:

Therefore, the total number is 13266257 + 36304935 + 6608040 + 17438748 = 73617980.
Back to table

5.5.4 3 cards totalling 23

By symmetry this is the same number as 3 cards totalling 4, which is 73617980.
Back to table

5.5.5 Skip straight

A skip straight is a 4 card straight where there’s a 1 number gap between each card: i.e. 0 2 4 6, 1 3 5 7, etc.
We break it down by the length of the skip straight:

So, our total is 2 1935495 + 2 5452920 + 2 10558836 = 35894502.
Back to table

5.5.6 Mixed straight

A mixed straight is a 4 card straight in which each of the cards is a different suit. I’m open to suggestions on how to count this efficiently.
Back to table

6 About this project

This page was generated from the LATEXsource with TeX4ht.