The last banana: A thought experiment in probability – Leonardo Barichello

The last banana: A thought experiment in probability – Leonardo Barichello

You and a fellow castaway
are stranded on a desert island playing dice for the last banana. You’ve agreed on these rules: You’ll roll two dice, and if the biggest number
is one, two, three or four, player one wins. If the biggest number is five or six,
player two wins. Let’s try twice more. Here, player one wins, and here it’s player two. So who do you want to be? At first glance, it may seem
like player one has the advantage since she’ll win if any one
of four numbers is the highest, but actually, player two has an approximately
56% chance of winning each match. One way to see that is to list all
the possible combinations you could get by rolling two dice, and then count up
the ones that each player wins. These are the possibilities
for the yellow die. These are the possibilities
for the blue die. Each cell in the chart shows a possible
combination when you roll both dice. If you roll a four and then a five, we’ll mark a player two
victory in this cell. A three and a one gives
player one a victory here. There are 36 possible combinations, each with exactly the same
chance of happening. Mathematicians call these
equiprobable events. Now we can see why
the first glance was wrong. Even though player one
has four winning numbers, and player two only has two, the chance of each number
being the greatest is not the same. There is only a one in 36 chance
that one will be the highest number. But there’s an 11 in 36 chance
that six will be the highest. So if any of these
combinations are rolled, player one will win. And if any of these
combinations are rolled, player two will win. Out of the 36 possible combinations, 16 give the victory to player one,
and 20 give player two the win. You could think about it this way, too. The only way player one can win is if both dice show
a one, two, three or four. A five or six would mean
a win for player two. The chance of one die showing one, two,
three or four is four out of six. The result of each die roll
is independent from the other. And you can calculate the joint
probability of independent events by multiplying their probabilities. So the chance of getting a one, two,
three or four on both dice is 4/6 times 4/6, or 16/36. Because someone has to win, the chance of player two winning
is 36/36 minus 16/36, or 20/36. Those are the exact same probabilities
we got by making our table. But this doesn’t mean
that player two will win, or even that if you played 36 games
as player two, you’d win 20 of them. That’s why events like dice rolling
are called random. Even though you can calculate
the theoretical probability of each outcome, you might not get the expected results
if you examine just a few events. But if you repeat those random events
many, many, many times, the frequency of a specific outcome,
like a player two win, will approach its theoretical probability, that value we got by writing down
all the possibilities and counting up the ones for each outcome. So, if you sat on that desert island
playing dice forever, player two would eventually
win 56% of the games, and player one would win 44%. But by then, of course, the banana
would be long gone.

100 thoughts on “The last banana: A thought experiment in probability – Leonardo Barichello”

  1. The chance for player 2 to win = the chance one dice is 5 or 6 (1/3) + 1/3 of the remaining 2/3 for the other dice being 5 or 6 = 5/9
    That's how I did it, it's just more confusing to explain.

  2. If you make it so player 1 wins if the highest number is 1, 2, 3, or 5 and player to wins if it's 4 or 6, they'll each have a 50% chance of winning.

    Also, if you use dice with sides that are divisible by 4, you can give the lower 4th and the upper 4th to one player and the middle half to the other and they'll both have a 50% chance of winning.

  3. Jesus Christ on a bike!

    ‘1-in-36 chance that ‘1’ will be the highest number’.

    But, you’re rolling two dice, so the minimum amount you could get would have to be 2, right?

  4. But how could they play game while they are hungry,they have no power so theoretically they both will is great to fight for a minute and there is 75% chances for the winning of cat practically.

  5. My Name Is Secret

    At first i didnt understand But then an idea popped into my mind So … think of it this way if 1,2,3,4 pop only one time player 2 Wins So that means 1,2,3,4 neted to pop 2 Times to give player 1 a victory.if 5 or 6 pops one time player 2 wins

  6. Lucky The Wolf Dog

    I am so confused. Not by the video, but by the comments. Some say "one person eats the other and banana". Others are saying "the cat doesn't eat bananas". And 50% of the comments are arguing to split the banana. Still, others are stuck how they got the dice in the first place, or why there's only 1 banana. Or why they can't go fishing. (If this comment seemed rude, then no offense to anyone.)

  7. But but……the chance of getting 5 or 6 is 2/6. 2/6 multiply by 2/6 is 4/36. So the chance of player 2 to win is 4/36 or 1/9 which is higher than player 1(16/36). Am i correct? Or is it? I need to know where is my mistake…

  8. This is beautiful. I especially like the visual showing the probability as a two dimensional square. If we had three dice you could show it as a cube.
    So what is the probability of four dice? That would be the probability of a fourth dimensional object, no?

  9. I thought the probability for player 2 winning the banana was 2/3

    Player two needs a roll of 5 or 6 to win, meaning he has a 1/3 chance to win

    But since two dice are rolled, the chance for player 2 to win is 1/3 + 1/3 = 2/3

  10. To all these smartass people saying that they would split the banana- You are not funny
    Its just a thought experiment, not a real life scenario

  11. Smack with this video every BDO player that says stuff like: "Fs are memes, I once got tet with 10 and failed DUO with 70"

  12. A different Thought Experiment helping you to understand the greatest enigma of all times. Who or what opened the Space-Time Inversion and set off the "Big Bang". Listen to my Podcast:

  13. I figured out easier by checking the cons of Player 1 and pros of Player 2:
    * Player 1 wins ONLY if both dices don't roll 5 or 6.
    * Player 2 wins if either dice rolls 5 or 6.
    * Player 1 needs both dices' results.
    * Player 2 doesn't.
    * Player 2 has "two tries" to win and Player 1 "one try".

  14. That moment when you study statistical physics and forget that normal people don't actually know this stuff.

  15. I misread the title as "The singing banana: A thought experiment in probability". I was hugely disappointed in the lack of James Grime in the video.

  16. Give the other castaway the banana, when their guard is down while eating, hit them with a big rock and feast on their corpse

    That's called survival

  17. Supposing the chance change if one player were to learn how to intuitively spin a die to get their desired outcome
    Fatalistic outcome obervance, all else is character

  18. Soo satisfying to watch these videos after you learn it's topic. Just had some lessons in college about probability and that's really accurate!

  19. But I've always thought that the number 1 side is slightly heavier because they indent holes into the surface for the number of spots, if this is the case number 6 (the opposite side) would come up more frequently than its 'probability' would claim?

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