The Gates to Heaven and Hell | Solution
The answer: It could take forever but probably will be less than three to six days.
We will want to know the probability that our man gets to heaven after a given number of days, i.e:
Pr(gets to heaven right away)
Pr(gets to heaven in under 2 days)
Pr(gets to heaven in under 3 days)
We know he is choosing the same door at random each time, so he has a ⅓ probability of picking either one each time he returns to the gate.
Pr(event 1, followed by event 2) = Pr(event 1) x Pr(event 2)
If our man picks the door to a one-day stay in hell, then he picks the door to heaven the second time, the probability of that would be:
Pr(picks the door to hell in round 1) x Pr(picks the door to heaven the second time) = ⅓ x ⅓ = 0.1111 = 11.11% chance
Pr(a given event) = Pr(one way to get there) + Pr(another way to get there) + …
If we want to know the probability that the man gets to heaven in 2 days or less, we need to add up the probabilities for all the different ways that could happen:
It takes 2 days.
Round 1, hell 1 day; round 2, hell 1 day; round 3, heaven
Round 1, hell 2 days; round 2, heaven
It takes 1 day.
Round 1, hell 1 day; round 2, heaven
It takes 0 days.
Round 1, heaven
We would add up all these probabilities:
Pr(heaven in 2 days or less)
= Pr(hell 1 day) x Pr (hell 1 day) x Pr (heaven)
+ Pr(hell 2 days) x Pr(heaven)
+ Pr(hell 1 day) x Pr(heaven)
= (⅓ x ⅓ x ⅓) + 2(⅓ x ⅓) + ⅓ = 0.59259 = 59.259% chance
So, there’s more than a 50% chance he will get to heaven in under 3 days.
To find out how long until he gets to heaven, we keep going in this fashion. There’s a little trick that makes this easy:
Pr(gets into heaven in under 4 days)
= Pr(gets in at 3 days) + Pr(gets in at 2 or fewer days)
In other words, we can use our previous answer, and we need ONLY calculate all the ways it would take him to get into heaven in exactly 3 days.
For Pr(gets in after 3 days), to save space, just write a number for the number of days in hell:
1,2, (gets in)
2,1, (gets in)
1,1,1, (gets in)
The probability, then, is:
2(⅓ x ⅓) + (⅓ x ⅓ x ⅓) = 25.956%
So, the probability, then, that he gets in after 3 days or less:
Pr(2 days or less) + Pr(3 days)
= 59.259% + 25.956% = 85.215%
So, most likely, he’s relaxing in paradise in 3 days’ time.
But possibly not … This is a beautiful problem that boils down to how many ways you can arrange groups of the numbers 1 and 2 to add them up to a given amount. For instance:
To compute all the ways it might take 4 days before he’s in heaven, you look at the following combinations:
1,1,1,1, (heaven—phew, enough of the heat already!)
This means: Pr(gets in after 4 days) = (⅓)5 + 3(⅓)4 + (⅓)3
And … the probability that he’ll be in heaven after 4 days’ time is: 93.033%.
It’s no guarantee, but for each day, the likelihood increases, to the point where he’s more likely to have been kicked to death by a donkey than to still be waiting in hell.
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