Simultaneous Probability There are three drivers and two roads, X and Y. The probability of choosing road X is p for all drivers, and the probability of choosing road Y is (1-p) for all drivers. To calculate the probability that all drivers choose road X simultaneous is p^3. However, what would 3p imply? What does it mean intuitively?
 A: That would be the expected number of drivers on road X.
Let $Z_i$ be an indicator for whether driver $i$ chose road X, i.e. $Z_i$ is one when driver $i$ chose road X and zero otherwise.
The expected number of drivers on X is $E\left[\sum_{i=1}^3 \, Z_i\right]$, which is $3\, E\left[Z_1\right]$ because everyone is identical and by linearity of expectation, and $E\left[Z_1\right] = p$ because the $Z$ are Bernoulli.
Intuitively, if p=1, you will have 3 drivers on road X.  If p=0.5, you will on average have 1.5. Makes sense right?
PS  It's possible that the decisions to drive on road X are not independent, in which case the probability of all three people choosing that road could be different from $p^3$.  For example, you could keep the same marginal distributions as in your setup, but assume that everyone does the same thing. For example, Mr 1 chooses road X wp p (and road Y wp (1-p)); and, conditional on that decision, Mr 2 and 3 do the exact same thing. Even in that case the expected number of people on road X would still be 3p.
