# expected value of Consecutive numbers [duplicate]

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$$\in\mathit S$$.

• Hint: What is the probability that both $1$ and $2$ are in your randomly chosen subset? – JMoravitz Dec 8 '18 at 16:52
• 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 – MathStudent Dec 8 '18 at 17:17

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 $$\frac{25×24×99}{100×99}=6$$.