Expectation value vs most probable value of radius in ground state of hydrogen What is the difference when we calculating expectation value for example in hydrogen atom in the ground state? The expectation value is 1.5*Bohr radius while the probability density has a max at Bohr radius. Does expectation value takes into account many measurements?
 A: Suppose you measure something with outcome “1” four times, outcome “2” once, and outcome “3” once.  The most probable outcome is “1” but the average is 9/6=1.16.  The most probable outcome need not be the average, and indeed the average may not even be a possible outcome.
A: The mode, median, and mean are three key statistics that all contend for qualifying as the "average" of a probability distribution. The mode is the point at which the probability density function (PDF) reaches its maximum value. The median is the point at which the probability of a value being less or greater are equal (in other words, it's where the cumulative probability density function (CDF) has a value of 1/2). The mean, or expected value, or first moment, is the weighted average (or $\int x f(x) dx$, where $f(x)$ is the PDF.
There are only a few probability distributions where the mode, median, and mean are equal to one another. The probability distribution that describes the distance between the center of a hydrogen atom and the electron in its ground state is one of many probability distributions where the mode, median, and mean differ.
In this case, the probability distribution function starts at zero at a zero distance, quickly rises to its maximum value (the mode) at the Bohr radius, and then gradually falls to zero as distance increases. It should not be surprising that in this case the mode is less than the median which in turn is less than the mean.
