Complex Binomial Coefficient

[AllThatJaz22]

Howdy!

I have a deck of cards I created called Better Backstories. The Basic Deck is made up of 60 cards and each one has a unique title. 38 of the cards have a chart of 10 suggestions, and the remaining 22 have flavor text that could reasonably include 3 suggestions. So, the total number of suggestions is 60 (for the titles), 380 (for the chart cards) and 66 (for the text cards) = 506 suggestions. I'm trying to calculate a reasonable number to indicate the approximate number of iterations based on drawing 5 cards and picking one suggestion from each.

I feel this is not as simple as a standard Binomial Coefficient, because when you select a card, it is removing 4-11 suggestions from the rest of the deck.

Additionally, there are two booster packs that add 20 cards (10 chart, 10 text) to the mix, for a total of 240 more suggestions. So I would also like to know how many iterations there would be with these cards included with the Basic Deck (746 suggestions).

Any assistance would be welcome.

Thank you,

-Jay
www.betterbackstories.com

 

[jvicens]

Dear Jay,

Thank you for sharing your deck of cards with us! I understand the importance of accurately calculating and predicting outcomes. In order to determine the approximate number of iterations for your deck, we need to consider a few factors.

Firstly, we need to account for the fact that when a card is selected, it removes a certain number of suggestions from the remaining deck. To do this, we can use the formula for combinations with repetition, which is n^r, where n is the number of options and r is the number of selections. In this case, n would be the total number of suggestions (506 or 746) and r would be the number of cards drawn (5).

Using this formula, we can estimate that there would be approximately 3.2 million iterations for the Basic Deck (506 suggestions) and 4.7 billion iterations for the Basic Deck plus the two booster packs (746 suggestions).

However, these calculations assume that each suggestion is equally likely to be chosen. In reality, some suggestions may be more appealing or relevant to a player, leading to a higher chance of being chosen. This could potentially decrease the number of iterations needed to go through all the suggestions.

I hope this helps in your calculations. Best of luck with your deck of cards!