31.1. Introduction http://www.ck12.org
bears repeating:
Probability of a particular end locationP(End Location) =∑Probabilities of outcomes that lead to that location
If all such outcomes are equally likely, then
P(End Location) = (Probability of a single outcome)×(Number of outcomes)Therefore, the probability of ending one step right is:
1 / 8 + 1 / 8 + 1 / 8 = 3 × 1 / 8 = 3 / 8
This reasoning allows us to find the probabilities of the other possible end locations as well, noted below the
arrows on the graph above. A grouping of possible end locations and their respective likelihoods is an example
of a probability mass function. Let’s define this important concept.
Probability Mass Function
A list of events with associated probabilities.In more mathematical language, we can represent of a set of events as
(Event 1 ,Event 2 ,Event 3 ,... ,Eventi).
The associated probabilities, meanwhile, are written as
(P(Event 1 ),P(Event 2 ),P(Event 3 ),... ,P(Eventi)).
In our example, the end locations of the three step random walk described above will can be written as
( 3 R, 1 R, 1 L, 3 L)
and their probabilities as
( 1 / 8 , 3 / 8 , 3 / 8 , 1 / 8 )
.
We can plot this distribution on a graph where final displacement (alternatively, number of steps — the two quantities
will be equal if we set the step lengths to 1, which we can with no loss of generality) from the origin is on the x-axis,
while its probability is on the y-axis: