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Each of the nine words in the sentence $\text{"The quick brown fox jumps over the lazy dog”}$ is written on a separate piece of paper. These nine pieces of paper are kept in a box. One of the pieces is drawn at random from the box. The $\text{expected}$ length of the word drawn is _____________. (The answer should be rounded to one decimal place.)
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ANS is 3.9

Each of the nine words have equal probability. So, expected length
$= 3 \times \frac{1}{9} + 5 \times \frac{1}{9} + 5 \times \frac{1}{9} + 3 \times \frac{1}{9} + 5 \times \frac{1}{9} + 4 \times \frac{1}{9}+ 3 \times \frac{1}{9}+ 4 \times \frac{1}{9}+ 3 \times \frac{1}{9}$
$= \frac{35}{9}$
$=3.9$

edited

let X be random variable denoting length of word drawn. We have to calculate E(X)

there are 4 words of length 3, 2 words of length 4 and 3 words of length 5

probability of choosing a word of length 3 = 4/9

probability of choosing a word of length 4 = 2/9

probability of choosing a word of length 5 = 3/9

E(X) = (4/9)x3 + (2/9)x4 + (3/9)x5 = 3.9

What question is this?

Count the letters and divide by the no. of words! 35/9=3.88888=3.9! ;)

0
thiz is random variable ques
+1 vote
There are total 9 words in the sentence. To find expected length first we count total length of the sentence i.e 35. Now to find expected length divide total length by total words i.e 35/9 so answer is 3.8 or 3.9

Another Explanation: Expected value = ∑( x * P(x) ) = 3*4/9 + 4*2/9 + 5*3/9 = 35/9 = 3.9
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