Question

What inferences about probability can I glean from a box and whisker plot?

Answer 1

A box and whisker plot should tell you the median value of your dataset, the maximum and minimum values, the range in which `50%` of values fall and the values of any outliers.

Explanation:

More technically, you can regard a box and whisker plot in terms of quartiles.

The top whisker is the maximum value, the bottom whisker the minimum value (assuming neither of the values are outliers (see below)).

Information on probabilities is gleaned from the positions of quartiles.
The top of the box is `Q1`, the first quartile. `25%` of values lie below `Q1`.
Somewhere inside the box will be `Q2`. `50%` of values lie below `Q2`. `Q2` is the median of the dataset.
The bottom of the box is `Q3`. `75%` of values lie below `Q3`.

`Q3 - Q1` (the length of the box) is the interquartile range, in which `50%` of values lie.

If a value falls above `Q3 + 1.5 (\text{IQR})` or below `Q1 - 1.5 (\text{IQR})`, it is classified a suspected outlier and will be marked with a circle on the box and whisker plot. If it falls above `Q3 + 3 (\text{IQR})` or below `Q1 - 3 (\text{IQR})` it is classified an outlier and marked with a solid circle.

For examples, see

and

These images are from this descriptive, useful page that you should read for further explanation and more examples.

These Wikipedia pages on quartiles, interquartile range and box and whisker plots should be useful also
Quartiles
Interquartile range
Box and whisker plots