A bag contains 26 tiles, each marked with a different letter of the alphabet. What is the probability of being able to spell the word math with four randomly selected tiles that are taken from the bag all at the same time?

Jun 3, 2017

 1/(""_26C_4).

Explanation:

There are $26$ tiles in the bag.

To spell the word MATH, w ways.e have toselect $4$ tiles from

the bag.

This can be done in ""_26 C _4 ways.

Out of this, the word MATH can be correctly spelled in only $1$ way.

Hence, the Prob. =1/(""_26C_4).

Jun 3, 2017

The probability is (4! 22!)/(26!), or about 0.00669%.

Explanation:

Drawing four tiles from the bag at one time is mathematically equivalent to drawing one tile 4 times, without replacement.

On our first draw, we have to select either M, A, T, or H from the alphabet bag in order to have a chance at spelling "MATH". Thus, on draw 1, there are 4 draws out of the possible 26 that we will call a "success".

If we get one of these 4 letters on draw 1, then on draw 2, we'll have 3 possible successful draws out of the 25 tiles left.

This pattern continues down to draw 4. Since each draw is independent, we just multiply the probabilities together:

$\frac{4}{26} \times \frac{3}{25} \times \frac{2}{24} \times \frac{1}{23}$

This is equal to

(4!)/(26!//22!)

which is the same as 1/(""_26C_4).