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I've been playing some Runescape lately and while training the "Thievery" skill, I encountered a little math question I would like to figure out:

The mechanics are simple: One after the other, the player picks up each fruit at random. If there are baskets in the player's inventory, certain fruits can be placed in them. Given the drop chances for each fruit and the rules below, what is the ideal number of empty baskets for the player to have in their inventory to maximize the number of fruits that can be picked up before the inventory is full?

Rules:

  • An empty inventory has 28 item slots.
  • Each fruit and each basket use up one item slot (regardless of type or contents).
  • Placing a fruit into a basket frees up an item slot in the inventory.
  • A fruit has to go into the inventory first, and can only then be placed into a basket. As long as the inventory is full, no more fruits can be picked up. (This is the point where the 'game' ends, unless one fruit can be moved into a basket.)
  • Dropping fruit is not allowed. Once it is inside the inventory it can only be moved into a basket (if possible) or has to remain inside the inventory.
  • Each basket can hold up to 5 fruits of the same type. Mixing different fruits in one basket is not possible.
  • A basket can only hold apples, bananas or strawberries.
  • Each fruit has a different drop chance (see table below).

Drop Chances:

Fruit Chance Basket
Cooking Apple 38% Yes
Banana 17% Yes
Jangerberries 9% No
Lemon 8% No
Redberries 7% No
Pineapple 6% No
Lime 5% No
Strawberry 4% Yes
Strange Fruit 3% No
Golovanova Fruit Top 2% No
Papaya Fruit 1% No

And here, a little visualization:

enter image description here

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It works best to start with a number of picked fruit and work back to the best number of baskets for that many fruit.

If you pick 50 fruit, you would, on average, have 19 apples, 8.5 bananas, 2 strawberries, and 20.5 other fruit. The apples would fit in 4 baskets, the bananas in 2, and the strawberries in 1.

If you have 7 baskets, that would leave 21 other slots for other fruit. This is almost exactly the number of "other" fruit you'd expect to have if you picked 50 fruit. Adding another basket would only help if you picked at least 22 apples, 12 bananas, or 7 strawberries. It is more likely that the extra basket would be empty or have only 1 fruit in it when you ran out of room because you picked up another "other" fruit. Thus, adding an eighth basket is unlikely to increase the number of fruit you can pick.

If you did have 7 baskets and picked 50 fruit, and hit exactly average, you'd have about a 50% chance of having one slot empty. You would also have about a 60% chance that the next fruit would be an apple, banana, or strawberry, and after you picked it up, you could move it into a basket and pick up yet another. So, I conclude the average number of fruit you could pick with 7 baskets would be about 51, with even odds that you'd max out at 50 or 52, instead.

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  • $\begingroup$ It seems to be correct, I'll give it a more in-depth look this weekend. But your approach makes sense :) $\endgroup$
    – Katai
    May 15 '20 at 13:06
  • $\begingroup$ Note that the maximum amount of fruits (if you are very lucky) would be 7*5+21=56 $\endgroup$
    – justhalf
    May 23 '20 at 3:45

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