### Brain Teasers

# Hats and Feathers

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

You have five boxes with five hats in each and each hat has five feathers with five colored dots. Now the hats are in a rainbow of color and so are the feathers but the dots are all various random combinations of red, yellow, and blue-green.

What is the probability of getting a hat with only red dots?

What is the probability of getting a hat with only red dots?

### Hint

There are only 3 colors, blue-green is a single color.### Answer

ONE CHANCE in 33,891,544,377.72How did I get that answer?

The first thing we have to do is figure out how many different combinations of dots there are on a single feather. Since there are three colors possible for any given dot and five dots on a feather, you would multiply 3 to the fifth power. 3x3x3x3x3=243

So, you have a 1 in 243 possibility of a feather having solid red dots. You have 5 feathers on one hat, so you must take 1/243 to the 5th power.

1/243 x 1/243 x 1/243 x 1/243 x 1/243 = 1/847,288,609,443.

Now, working with the fact that you have 25 hats to work with (five boxes with five hats each), you must divide the 847+ billion chances by 25, or 847,288,609,443/25, which leaves you with one chance in 33,891,544,377.72.

In other words, not very likely.

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## Comments

I can't put my finger one it but i think you did something wrong here.......

why do the odds get worse when more hats are added? Shouldn't it be 25/243 because with one hat it's only one in 234 but now you get 25 chances?

I would think the number of dots of each color would matter. It would change the odds dramatically if, for example, there were 10 red dots out of the total 625. Since you can't have an equal number for each color (625/3=208.3) you need to know how big a pool you're pulling from.

Ok, the color of the dots on the feather are a random variable confined to 3 colors. There are no limitations placed on the colors choosen. If you have 5 dots with a posibility of 3 colors then you multiply 3 to the 5th power to get 246 different combinations. I could list them all out here but for the sake of brevity I'll skip it. Now we Know that the odds of getting one feather with nothing but red dots is 1 in 246. The next element of the puzzle is figuring out the odds of getting 5 feathers just alike on the same hat. So we multiply 1/246 to the 5th power giving us 844+ billion combinations. And since we have 25 hats, we have 25 chances to get that specific hat so we divide the 844+ billion odds by 25 and get 1 in 33+ billion odds of getting that hat. It's really a simple probability problem it just has big numbers.

A=NUMBER OF DOTS

B=NUMBER OF COLORS

C=NUMBER OF FEATHERS

D=NUMBER OF HATS

E=NUMBER OF DOT POSIBILITIES

F=ODDS OF GETTING 5 LIKE FEATHERS

B^A = E

E^C = F

D/F = THE ANSWER

A=NUMBER OF DOTS

B=NUMBER OF COLORS

C=NUMBER OF FEATHERS

D=NUMBER OF HATS

E=NUMBER OF DOT POSIBILITIES

F=ODDS OF GETTING 5 LIKE FEATHERS

B^A = E

E^C = F

D/F = THE ANSWER

1 in 34 billion????

25 hats is a lot to choose from

25 hats is a lot to choose from

The answer given is incorrect. The mistake is in the assumption that the chance that one of the 25 hats is all red is exactly 25 times as likely as that any given hat will be all red. If this were true, then by the time you had 33.9 billion hats, the probability of an all red hat would be one. But clearly it's possible to have arbitrarily many hats with none of them all red. The correct method is to find the probability that any given hat ISN'T all red. That's 1 - (1/243)^5. Then take this probability to the 25th power, and you get the probability that none of the first 25 hats are all red. One minus this probability is the probability that one of any group of 25 hats IS all red. So the answer is 1 - (1 - (1/243)^5)^25, or one chance in 33,890,317,956.94. Close to the answer given, but not the same.

I agree with Bender. Rayneeday' s solution is slightly flawed but it doesn't matter when the numbers are so small.

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