Question

A bag contains 26 Scrabble tiles, one for each letter of the alphabet. We draw 4 tiles from the bag at random, without replacing the tiles after they are drawn. a) If we write down the letters in the order in which they are drawn, what is the probability that they will spell the word "MATH"? b) What is the probability that they will spell either "MATH" or an anagram of the same word? ex. "HTAM" c) what is the probability that our letters will be "AEIO," in that order? d) What is the probability that the letters will all be vowels? (Count y as a vowel)

Answer 1

Let's look case by case.
a)
You first have to draw the M tile, then the A tile, then the T tile, and then the H tile. What are the probabilities of each of these?

Answer 2

M Tile is 1/26? A is 1/25...?

Answer 3

Yes.

Answer 4

ok!!

Answer 5

b) The order doesn't matter. The first tile can be M,A,T, or H. The second tile any of the remaining three, and so on. What are the probabilities for each of these steps?

Answer 6

Okay so...the probability for the first tile...could be M, A, T, H or any other letter in the bag is 1/26. Next is 1/25, next is 1/24 and next is 1/23... so 26. 25. 24. 23

Answer 7

Close.
How many favorable options are there for the first tile?

Answer 8

OH...4.. 4/26?

Answer 9

Yup

Answer 10

What about the second tile you draw?

Answer 11

3/25?

Answer 12

Yes

Answer 13

woo!!

Answer 14

Okay so i get the pattern now but how can i answer the question? :)

Answer 15

So, for part b we have:
\[\frac{ 4 }{ 26 } \times \frac{ 3 }{ 25 } \times \frac{ 2 }{ 24 } \times \frac{ 1 }{ 23 } = \frac{24}{358800} = \frac{1}{14950}\]

Answer 16

Okay i see! So the probability that we draw the tiles in order of spelling out "MATH" is 1 in 14950!

Answer 17

That's for out of order. For in order, it would be 1 in 358800 (The initial numerator is 1 but the denominator stays the same)

Answer 18

which is the answer to the next question.. and 'anagram' of the same word!

Answer 19

Oh. Sorry for in order (b) it would be 1 in 358800 chance. In an anagram situation..(c) it would be 1 in 14950 chance

Answer 20

Yup, except in order is (a) and anagram is (b), unless I'm reading the problem wrong.

Does this make sense? For the anagram part, there are more possible ways to draw tiles that work than when you need a specific number, so there's a higher probability of it working.

Answer 21

Okay, no you are reading correctly, im just confusing things. Thank you i understand those now!

Answer 22

Let's look at part (c). It should be very close to part (a)

Answer 23

Yes, just reading the question over again i can tell. So for the letter A. I know that the probability of me drawing it is 1 in 26. For E it is 2 in 25...but because i have to figure out the probability of drawing AEIO in order it would be exactly as if i were drawing MATH in that order....with 4 letters and all...so the probability would be 1 in 358800 wouldnt it?

Answer 24

Yup.

Answer 25

And now for part d, where you can draw any four vowels. (This is similar, but not quite the same as, part b)

Answer 26

okay so this time there are 6 possible letters being drawn...A E I O U Y...4 letters are drawn... the possibility of drawing A (or E, I, O, U, or Y) is 6 in 26?

Answer 27

Yes

Answer 28

Yay!! Thank you so much.

Answer 29

What about the second tile?

Answer 30

the second tile would be 5/25, 4th tile 4/24, 3rd tile 3/23, 2/22, and 1/21

Answer 31

But remember, you are only grabbing 4 tiles.

Answer 32

\[\frac{ 6 }{ 26 } \times \frac{ 5 }{ 25 } \times \frac{ 4 }{ 24 } \times \frac{ 3 }{ 23 } = \frac{ 360 }{ 358800 } = \frac{ 3 }{ 2990 }\]

Answer 33

oh right...okay...so

Answer 34

im confused again

Answer 35

i understand the fact we have 6 vowels and if i can only draw 4 tiles..that makes the chances greater

Answer 36

then drawing something in order.

Answer 37

To find the probability of grabbing 4 vowel tiles, we only need to worry about the first 4 tiles we grab. We don't care what the fifth or sixth tiles are.

Answer 38

Yes thats right. Oh so i just continue the fraction of 3/2990?

Answer 39

3/2990 is already simplified, so it's a 3 out of 2990 chance of the first 4 tiles you grab being vowels.

Answer 40

Gotcha...okay! great. I really appreciate you taking the time. Thanks again!

Answer 41

No problem!