what is the probability of drawing 4 tiles that spell MATH out of 26 tiles containing the whole alphabet?

Question

i have 26 letter tiles a-z

?probability of drawing 4 letters to spell MATH

Answer 1

There are two ways to solve this problem, both of which will give the same answer.

1. The first way is to use a combination to find the total number of ways to draw any four tiles. This is the simpler option if you have a calculator that can do combinations for you. I chose to do a combination instead of a permutation because you can just rearrange the letters after you draw them to spell the word. If you had to draw the letters all in order, you would use a permutation.

 First, find C(26, 4)

  C(26,4)=26!/(4!*(26-4)!)=14950

The combination gives us the total number of ways of selecting any 4 tiles in any order, and there is 1 way of drawing the correct tiles in any order. This means we divide 1 by C(26, 4). Our answer is

P=1/14950 (or 0.00006689)

 

2. Our second option is to just multiply the probabilities of drawing the correct tile for each of the 4 drawings. (You don't have to assume that you draw them one at a time for this method to work.) When we draw the first tile, it can be any one of the 4 letters we need, so there is a 4/26 probability of getting a letter we need on the first draw. On the second draw, it can be any one of the 3 letters we still need, but now out of 25 total tiles because we already removed the first one. We do this for the 4 drawings, which looks like

P=(4/26)*(3/25)*(2/24)*(1/23)=24/358,800. This simplifies down to our previous answer of P=1/14950.

 

These two methods do the same thing, but the first option just uses the combination without really providing much insight into what is actually happening with the probabilities.

Answer 2

n = 26 letters in the alphabet

r = 4 letters to select

 

First, I use permutation without repetition to figure out how many ways to arrange 4 letters to create the word MATH.

 

P(n, r) = P(26, 4) = 26!/(26 - 4)! = 26!/22! = 26*25*24*23 = 358,800

 

Second, I use permutation with repetition to figure out how many ways to arrange letters to create a 4-letter word.

 

n^r = 26⁴ = 26*26*26*26 = 456,976

 

Finally, lets put those two together as a probability.

 

P(4 letters that spell MATH) = P(n, r)/n^r = 358,800/456,976 = 0.7852

 

The probability of drawing 4 letters that spell MATH out of the 26 letters in the alphabet is 0.7852.