Page 1 of Brain teaser/puzzle...

Came across this today, what do you think?....

A monkey is sat at a typewriter that has only 26 keys, one per letter of the alphabet. The monkey types at random, with a constant speed of one letter per second. It favours no letters: all letters at any second have a 1/26 probability of being typed.

Which of the following is greater?

a) the average time it will take the monkey to type “abracadabra”

b) the average time it will take the monkey to type “abracadabrx”

This is not a trick question. The word ‘abracadabra’ has 11 letters, and therefore has a probability of (1/26)^11 of appearing during any 11 second spell. Likewise, the word ‘abracadabrx’ has 11 letters, and also has a probability of (1/26^)11 of appearing during any 11 second spell. Yet this observation does not entail that they will occur on average after the same amount of time.

(The question is NOT asking which word the monkey will type first. The question is asking what will happen in the long run. Imagine that the monkey has been typing for such a long time that both ‘abracadabra’ and ‘abracadabrx’ have appeared many times; on average, how long did it it take the monkey to type each of these words?)
Another way of phrasing the question would be: over the long run, which of ‘abracadabra’ or ‘abracadabrx’ appears more frequently? The one that is more frequent is the one it takes, on average, less time to get to.

So, is it abracadabra, abracadabrx or is it the same?

Will credit the source after a few answers or so.

Ste

it should be the same shouldn't it?

like if you flip a coin 100 times, it should land heads 50% of the time, it won't, but should do :)

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Quote:
admars says...
"it should be the same shouldn't it?"Possibly..

Ste

First off... I always get these wrong... anyway...

My first thought was, it's the same, otherwise it's the gamblers fallacy, that because an 'a' came up before an 'x' that is more likely when the probability is really the same...

...but... then I started to think about it from another angle. It's a chain of letters that resets, and part of the start of the word is a subset of the full word.

And now I am thinking, abracadabra can contain the start of both words, but abracadabrx does not. So, maybe this makes the following true?

1. when abracadabra appears first, abracadabrx can follow sooner:
abracadabracadabrx
2. when abracadabrx appears first, abracadabra cannot immediately follow so the closest it can appear in this case is:
abracadabrxabracadabra

Which makes the odds ever so slightly favouring abracadabrx over a very large sample size?


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This item was edited on Friday, 24th March 2023, 20:48

Not sure if this helps or hinders but...
FURTHER CLARIFICATION: If the monkey types ‘abracadabracadabra’ this only counts as one ‘abracadabra’. Likewise, ‘abracadabrabracadabra’or 'abracadabrabracadabrx' is only one ‘abracadabra’. In other words, you need to type the word ‘abracadabra’ completely, and that counts as one appearance, and then you need to type it completely again for the next appearance.
Although I think maybe a puzzle with extra clarifications isn't the best puzzle anyway?...

Ste

Quote:
sj says...
"Although I think maybe a puzzle with extra clarifications isn't the best puzzle anyway?..."
you mean a bit like when you explain a joke to someone, and they still look at you blankly?


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