Finally, victory points are the best thing to get. However, this is just a 16% chance. That means you would need to spend 18 resources on average (3 per development card, 6 cards at a 16% chance each) to get one victory point. Obviously, this is a far worse choice than just building those roads and settlement, or building that city.

Unfortunately, you've calculated the average resource expenditure wrong. It's actually 12 resources (4 cards) not 18 resources (6 cards) if no development cards are purchased by anyone else.

First of all, you seem to have added 16% until you got near 100%, while to calculate the AVERAGE expenditure you should calculate to near 50%.

The correct formula is slightly complicated for two reasons:

1. After each card you buy that ISN'T a victory point, the chance that the next one IS a victory point is HIGHER. 4/25 = 16%; 4/24 = 16.7%

2. You only have to buy another card (to get one victory point) if the previous one was NOT a victory point. This means that after the first card you can't just use the chance the card is a victory point -- you need to calculate conditional probability. So the chance you get a victory point on the second card is (chance first card WASN'T a victory point) * (chance second card IS a victory point).

Put these together and you get:

Card 1:

(4 / 25)

Card 2 if Card 1 wasn't a victory point:

(4 / 24) * (1 - 4 / 25)

Card 3 if neither Card 1 nor Card 2 were victory points:

(4 / 23) * (1 - 4 / 25) * (1 - 4 / 24)

Card 4 if none of the previous cards were victory points:

(4 / 22) * (1 - 4 / 25) * (1 - 4 / 24) * (1 - 4 / 23)

Card 1 + Card 2 + Card 3 + Card 4 = 52.7% chance of getting a victory point

So on average, you will have received a victory card after only 12 resources have been spent, assuming you're starting from the full deck and no one else purchases cards.

EDIT:

Another (approximate) way to think of this is to imagine if you divide the deck into four equal parts. Each part will have 6.25 cards: 1 victory point, 3 soldiers, and 2.25 other cards. In this case, since there is only 1 victory point, buying half the cards gives you a 50% chance of getting a victory point: 3.125 cards; round this up to 4.

The table below shows three things:

1. The raw probability: the chance that a given draw is a victory point if none before were;

2. The conditional probability: the chance that you draw your first victory point on a given draw;

3. The cumulative probability: the chance you draw a victory point on or before a given draw.

You can see that the 4th card purchased yields a cumulative probability of over 50%.

- Code: Select all
`card raw_prob cond_prob cum_prob`

1 16.00% 16.00% 16.00%

2 16.67% 14.00% 30.00%

3 17.39% 12.17% 42.17%

4 18.18% 10.51% 52.69%

5 19.05% 9.01% 61.70%

6 20.00% 7.66% 69.36%

7 21.05% 6.45% 75.81%

8 22.22% 5.38% 81.19%

9 23.53% 4.43% 85.61%

10 25.00% 3.60% 89.21%

11 26.67% 2.88% 92.09%

12 28.57% 2.26% 94.35%

13 30.77% 1.74% 96.09%

14 33.33% 1.30% 97.39%

15 36.36% 0.95% 98.34%

16 40.00% 0.66% 99.00%

17 44.44% 0.44% 99.45%

18 50.00% 0.28% 99.72%

19 57.14% 0.16% 99.88%

20 66.67% 0.08% 99.96%

21 80.00% 0.03% 99.99%

22 100.00% 0.01% 100.00%

I'd say any of the other development cards are worth at least 2 resources (even if you're not going for largest army), so spending 12 resources, and getting 3 other cards means you're really only spending (on average, from an untouched deck) 6 resources to get the victory point --- much more reasonable than the 18 you suggest it would cost in your article. It's the lost additional resources that would have resulted from investing in settlements and cities, rather than their cost per victory point, that make this strategy a loser (unless you're also going for largest army).