### Brain Teasers

# Prime Suspects

Probability
Probability puzzles require you to weigh all the possibilities and pick the most likely outcome.

Two hundred people in an auditorium are asked to think of a single digit number from 1 to 9 inclusive and write it down. All those who wrote down a prime number are now asked to leave. Ninety people remain behind in the hall. How many of these are expected to have written down an odd number?

### Hint

Remember that 1 is not a prime number.### Answer

Those that remain behind must have written {1,4,6,8,9} and from this only {1,9} are odd. The probability of an odd number is thus 2/5.Expected number of odds is 2/5 * 90 = 36.

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## Comments

they didn't all write a number grr... or is my math wrong

DOH, helps if you don't count 1 as a prime number! I came up with 1/4 * 90 instead of 2/5 * 90.

Nice and easy.

You shouldn't really count 1 as a prime number because its not.

Heres why...

The integers form a UFD, in a UFD primes and irreducibles are the same. Since units aren't irreducibles, and 1 is a unit, then 1 can't be prime.

You shouldn't really count 1 as a prime number because its not.

Heres why...

The integers form a UFD, in a UFD primes and irreducibles are the same. Since units aren't irreducibles, and 1 is a unit, then 1 can't be prime.

This assumes that each of the 9 numbers is equally likely to be chosen. Experimental evidence shows this to be false. On what did you base your assertion that it's true?

Assuming equal likeliness for digits 1-9 is the obvious thing to do, in practice you might find number 7 is chosen more because some ppl like the number 7.

But for sake of simplicity we can assume equal likeliness.

When tossing a coin, its not equally likely to get a head or tail, but it practically is, so we just call it 1/2.

But for sake of simplicity we can assume equal likeliness.

When tossing a coin, its not equally likely to get a head or tail, but it practically is, so we just call it 1/2.

The fact that only 90 people remain indicates that there is not equal probability of choosing any number. If there wereequal probability, 111 would remain. Knowing that, why would the answer be based on equal probability?

What experimental evidence Norcekri? How many trials? Is it culturally biased? Do you agree with Lelrod's assertion that there must be 111 if all selections are equally likely?

you guys are great and smart ! since the language questions are my forte , my answer to this one was ... who cares !

I also think that solution is not clear - it is based on 'equal probability' for all numbers, and yet out of 200 people only 90 remained, which is NOT expected 5/9=111 (ie primes 2,3,5,7 out, 5 of 9 remains in). It is much closer to 4/9=88, indicating that one more 'prime' was used, and only candidate is #1. That would result in 4,6,8,9 remaining, and 1/4*90= only 22 would have odd(#9)

Apr 06, 2006

Hey,

The problem is quite simple. Lemme explain...

The numbers which the remaining 90 people must have written are 1,4,6,8,9

now... the to odds here are 1 and 9... Probably of selecting them is 1/5 + 1/5 = 2/5

So the number of guys who might have written 1 or 9 is 2/5*90 = 36... Cool !

The problem is quite simple. Lemme explain...

The numbers which the remaining 90 people must have written are 1,4,6,8,9

now... the to odds here are 1 and 9... Probably of selecting them is 1/5 + 1/5 = 2/5

So the number of guys who might have written 1 or 9 is 2/5*90 = 36... Cool !

Well of course it didn't out to be exactly 111 that remained, that would be unlikely compared to something else happening. But I shouldn't think that would give you a better clue as to the distribution of the numbers left (1,4,6,8,9) so why not assume eaqually likely, seems better than guessing a random distribution.

Great teaser!!

Awsome teaser

Easy one.

To cms: Maybe a disproportionate number of people left because they liked 7 and it's prime. Nevertheless, they're gone now and no longer influencing the outcome.

To cms: Maybe a disproportionate number of people left because they liked 7 and it's prime. Nevertheless, they're gone now and no longer influencing the outcome.

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