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)

Etc.

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)

+ 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!)

1,1,2, (ditto)

1,2,1, (ditto)

2,1,1, (ditto)

2,2, (ditto)

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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