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We choose randomly a subset $\mathit S$ of size 25 from the set $\{ 1,2,...,100\}$ , what is the expected value of number of Consecutive numbers in $\mathit S$?

Consecutive numbers in $\mathit S$: is a pair $\{ i,i+1\}$ where i$\in$$\mathit S$ and i+1$\in$$\mathit S$.

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  • $\begingroup$ Hint: What is the probability that both $1$ and $2$ are in your randomly chosen subset? $\endgroup$ – JMoravitz Dec 8 '18 at 16:52
  • $\begingroup$ it is:$\frac{\binom{98}{23}}{\binom{100}{25}}$ as Parcly mentioned below,that is the probability of any consecutive pairs not only 1 and 2.If we multiply that by number of possible consecutive pairs we will get the expected value we are looking for.@JMoravitz $\endgroup$ – MathStudent Dec 8 '18 at 17:17
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Linearity of expectation is the key. There are 99 consecutive pairs, each one having a $\frac{\binom{98}{23}}{\binom{100}{25}}=\frac{25×24}{100×99}$ chance of being selected. Thus the expected number of consecutive pairs is $\frac{25×24×99}{100×99}=6$.

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