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## Hats and Feathers

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

 Puzzle ID: #3711 Fun: (2.21) Difficulty: (2.84) Category: Probability Submitted By: rayneeday Corrected By: cnmne

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?

ONE CHANCE in 33,891,544,377.72

How 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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