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I have a very simple question (I think). I was helping my kid with probability lessons provided on http://www.mathgoodies.com/worksheets/pdf/unit6_wks1_key.pdf. One of the questions, "What is the probability of choosing the letter i from the word probability?" has the answer 2/9 (following logic of having a sample set of 9 letters in "probability"). I contacted mathgoodies.com and suggested that if they follow that logic, then (a famous example, I guess) the probability of choosing S from MISSISSIPPI is 4/4. Which obviously isn't. Or am I confused?

Thank you

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  • $\begingroup$ Thank you, Clement. I was not confused after all. $\endgroup$ – Sasha Nov 4 '14 at 23:23
  • $\begingroup$ I actually had trouble understanding the question until I noticed that the last "probability" should have been in quotes, as it is about the word "probability" (a sequence of 11 letters), not about some unspecified word probability (the probability of some word(s)). Of course the mistake was already in the linked exercise sheet. $\endgroup$ – celtschk Jan 21 '17 at 11:04
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It depends on how they define "choosing from", I reckon, but indeed according to the natural interpretation (picking a letter uniformly at random), it should be $2/11$ The sample space is of size 9, indeed, but not all elements have equal probability:

  • p,r,o,a,l,t,y have each probability 1/11
  • b,i have each probability 2/11

The total is indeed $7\cdot \frac{1}{11}+2\cdot \frac{2}{11}=1$.

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