Cake
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    • @apm I think yours is a good example to show the general problem we're talking about, if expanded slightly.

      In that example
      > our minimum deck size is three
      > we're drawing a hand of two cards
      > drawing {duck, pizza} is a good event, drawing {X, cake} in our initial hand is bad, but we still want to draw cake at some point.

      If our deck consists of just {duck, pizza, cake}, then as you point out we have a probability of 1/3 to get the good event. We have a 2/3 probability of immediately drawing {X, cake} - but we only want cake later.

      Replacing cake with another pizza (deck is {duck, pizza, pizza}), we increase the "good draw" probability to 2/3, but at the same time we decrease our probability of drawing cake to 0 - and we want to eventually draw cake.

      So, if instead of replacing cake with pizza, we add another pizza on top of that - making our deck {duck, pizza, pizza, cake} - we will still have a 1/3 probability of drawing {duck, pizza}, while our probability of drawing cake early is reduced to 1/2 while still being available at some point.

      Adding another duck for {duck, duck, pizza, pizza, cake} increases our "good draw" probability to 2/5, while decreasing the probability for an early cake draw to also 2/5. On the other hand, the fact that we've added another card also means that there's a 1/5 probability of cake being the fifth card in our deck. If we're only ever drawing four cards, this can mean that we're not drawing cake at all

      So, in this toy example, depending on what value we want to give to the events "initial good draw", "eventual cake draw" and "no cake draw at all", it might be sensible (or not) to play with a deck consisting of four or five cards instead of just three.

      At the same time though, we're making pretty big assumptions again and stay very vague in some regards, so I'm not convinced that this outcome can just be generalized to any deck size and any value for synergistic effects between cards.