smc

There are $29$ possible pairs of consecutive integers, namely $p_1=\{1,2\},\cdots ,p_{29}=\{29,30\}$. Define a random variable $X_i$, with $X_i=1$, if $p_i$ is part of the 5-element subset, and $0$ otherwise. Then the number of pairs of consecutive integers in a $5$-element selection is given by the sum $X_1+\cdots +X_{29}$. By linearity of expectation, the expected value is equal to the sum of the $\mathbb{E}[X_i]$: $$\mathbb{E}[X_1+\cdots +X_{29}]=\mathbb{E}[X_1]+\cdots +\mathbb{E}[X_{29}]$$ To compute $\mathbb{E}[X_i]$, note that $X_i=1$ for a total of ${}_{28}{C}_3$ out of ${}_{30}{C}_5$ possible selections. Thus$$\mathbb{E}[X_i]=\frac{\left(\genfrac{}{}{0ex}{}{28}{3}\right)}{\left(\genfrac{}{}{0ex}{}{30}{5}\right)}=\frac{1}{29}\cdot \frac{2}{3},\text{which implies}\mathbb{E}[X_1+\cdots +X_{29}]=\boxed{\frac{2}{3}}.$$