Last Updated: May 24, 2019

# Red Card

## Introduction

Red Card is an "arcade game" by Gluck Games. It features a field of 49 soccer balls. As the player picks them, either a trophy or red card is revealed. A trophy wins and a red card loses.

## Rules

1. The game starts with a field of 49 soccer balls. Behind them are placed randomly six red cards and 43 trophies.
2. Play starts with the player making a bet.
3. Then the player chooses one of the soccer balls to scratch.
4. If the soccer ball reveals a red card, then the player loses.
5. If the soccer ball reveals a trophy, the player wins and his balance increases.
6. The win table is shown below, based on a starting wager of one unit.
7. After each successful turn, the player may quit and cash in or go back to rule 3 and scratch another soccer ball.

The following table shows what the player may cash in for after any number of turns from 1 to 26. Sorry, but the rule screen doesn't say the full pay table and 26 successful turns is as far as I ever got.

Turn Pays
1 1.1
2 1.2
3 1.4
4 1.6
5 1.8
6 2
7 2.5
8 3
9 3.5
10 4
11 4.5
12 5
13 6
14 7
15 8.5
16 10
17 12
18 14.5
19 17.5
20 20
21 25
22 32
23 40
24 50
25 65
26 85
27 ?
28 ?
29 ?
30 ?
31 ?
32 ?
33 ?
34 ?
35 ?
36 ?
37 ?
38 ?
39 ?
40 ?
41 ?
42 ?
43 ?

## Analysis

I view each soccer ball picked as a separate bet. That said, the following table shows the probability of winning and expected return of each individual pick. The third column, titled "win next pick," shows what each bet pays on a "for one" basis, relative to the amount won from the previous pick. For example, the second pick pays 1.2 for a bet of 1.1, or 1.090909. The table stops at a total win of 85, because I don't know what you win if you go beyond that.

### Red Card Analysis — Bet by Bet

Turn Total Win Win Next Pick Probability Return
1 1.1 1.100000 0.877551 0.965306
2 1.2 1.090909 0.767857 0.954545
3 1.4 1.166667 0.669833 1.017730
4 1.6 1.142857 0.582463 0.993789
5 1.8 1.125000 0.504802 0.975000
6 2 1.111111 0.435965 0.959596
7 2.5 1.250000 0.375133 1.075581
8 3 1.200000 0.321542 1.028571
9 3.5 1.166667 0.274487 0.995935
10 4 1.142857 0.233314 0.971429
11 4.5 1.125000 0.197420 0.951923
12 5 1.111111 0.166248 0.935673
13 6 1.200000 0.139289 1.005405
14 7 1.166667 0.116074 0.972222
15 8.5 1.214286 0.096176 1.006122
16 10 1.176471 0.079204 0.968858
17 12 1.200000 0.064803 0.981818
18 14.5 1.208333 0.052652 0.981771
19 17.5 1.206897 0.042462 0.973304
20 20 1.142857 0.033969 0.914286
21 25 1.250000 0.026941 0.991379
22 32 1.280000 0.021168 1.005714
23 40 1.250000 0.016464 0.972222
24 50 1.250000 0.012665 0.961538
25 65 1.300000 0.009625 0.988000
26 85 1.307692 0.007219 0.980769

The next table analyzes the overall return by all stopping points through the 26th turn. The third column shows the probability of surviving past that turn. The total bet shows the average amount you would have bet if it was the goal to survive to that turn and then quit. The final column shows the ratio of the expected win to the average amount bet to get to that win. In other words, how I would define the expected return of a goal of getting to that win.

The last column is maximized at 99.534% with a strategy of surviving past the 9th pick, or getting to a win of 3.5 or lose trying.

### Red Card Analysis — Bet by Bet

Turn Total Win Probability
Survival
Win Next Pick Return
1 1.1 0.877551 1.000000 0.965306
2 1.2 0.767857 1.965306 0.960021
3 1.4 0.669833 2.886735 0.978441
4 1.6 0.582463 3.824501 0.982205
5 1.8 0.504802 4.756442 0.980793
6 2 0.435965 5.665085 0.977393
7 2.5 0.375133 6.537015 0.990490
8 3 0.321542 7.474846 0.995268
9 3.5 0.274487 8.439473 0.995344
10 4 0.233314 9.400179 0.992900
11 4.5 0.197420 10.333436 0.989199
12 5 0.166248 11.221824 0.984962
13 6 0.139289 12.053065 0.986371
14 7 0.116074 12.888799 0.985454
15 8.5 0.096176 13.701319 0.986680
16 10 0.079204 14.518812 0.985676
17 12 0.064803 15.310848 0.985477
18 14.5 0.052652 16.088483 0.985298
19 17.5 0.042462 16.851942 0.984754
20 20 0.033969 17.595020 0.981778
21 25 0.026941 18.274405 0.982135
22 32 0.021168 18.947934 0.982973
23 40 0.016464 19.625311 0.982602
24 50 0.012665 20.283873 0.981918
25 65 0.009625 20.917105 0.982102
26 85 0.007219 21.542738 0.982064

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