# Probability of picking two letters from the word MATHEMATICAL [closed]

If you select 3 letters (with replacement) from the word MATHEMATICAL, what is the probability of getting two 'M's.

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## closed as off-topic by mathlove, PVAL, Kf-Sansoo, Claude Leibovici, Tunk-FeyAug 25 '14 at 11:50

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Thanks for the answers. I get 5/72 as well. But the correct answer is 11/144. Could some one please explain this answer to me. Thx. – Susie Aug 25 '14 at 11:08
I do not agree with the answer of 11 / 144. – bobbym Aug 25 '14 at 11:12
I tend to agree with you. But the answer in the text book is 11/144 – Susie Aug 25 '14 at 11:16
The textbook's answer is wrong. – David Mitra Aug 25 '14 at 11:21
Much appreciated. Thank you. – Susie Aug 25 '14 at 11:24

How many letters are there in the word MATHEMATICAL? $12$

How many letters are $M$ here? $2$

We want the probability of getting two 'M's takes considering all three cases. When we get a non-M on the first, second or third pick.

$\frac{10}{12}*\frac{2}{12}*\frac{2}{12}+\frac{2}{12}*\frac{10}{12}*\frac{2}{12}+\frac{2}{12}*\frac{2}{12}*\frac{10}{12}=30*\frac{2^2}{12^3}=\frac{5}{72}$

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The number of M's obtained is binomially distributed with $p={1\over6}$. The probability to obtain exactly two M's in three draws is therefore given by $${3\choose 2}\ \left({1\over6}\right)^2\ \left({5\over6}\right)^1={5\over72}\ .$$

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I am getting,

$$3\left ( \frac{2}{12}\right )\left ( \frac{2}{12}\right )\left ( \frac{10}{12}\right )=\frac{5}{72}$$

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Why do we have three answers? What is wrong with my method and/or $8\pi \mathrm{r}$'s? – Tony Aug 25 '14 at 11:05
I think because I am treating (m,m,X) as different than (X,m,m) and (m,X,m). – bobbym Aug 25 '14 at 11:09
Oh good point, thank you for that! – Tony Aug 25 '14 at 11:10
I am your upvote here, and I have no idea why I got one more upvote than you, considering my answer was obtained from your help! – Tony Aug 26 '14 at 14:42
It us not a big deal, glad you got them. Looks like they are closing the question anyway. – bobbym Aug 26 '14 at 15:08

Number of 'M's in MATHEMATICAL = 2

Number of letters in MATHEMATICAL = 12

Probability of getting an 'M': $\frac{2}{12}$

Probability of not getting an 'M': $\frac{10}{12}$

Ways to shuffle these three: $\binom{3}{2} = 3$ ($3$ letters of which $2$ are identical)

Probability: $$\left ( \frac{2}{12} \right )^2 \cdot \frac{10}{12} \cdot \binom{3}{2} = \frac{5}{72}$$

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