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stat quest hello I'm Josh Dahmer and
welcome to stat quest today we're gonna
be talking about the difference between
probability and likelihood these are two
closely related concepts that are very
easy to get confused even I mix them up
from time to time so enough of this
jibber-jabber let's get down to it for
me the easiest way to understand the
difference between probability and
likelihood is to just see it in pictures
so let's start by looking at probability
with respect to a normal distribution
keeping in mind that this concept
applies to all continuous distributions
in this case let's imagine that this is
a distribution of Mouse weights it has a
mean of 32 grams and a standard
deviation of 2.5 on the low end we have
24 grams and on the high end we have 40
grams the probability that we will weigh
a randomly selected Mouse between 32 and
34 grams is the area under the curve
between 32 and 34 grams in this case the
area under the curve equals zero point 2
9 meaning there's a 29% chance a
randomly selected Mouse will weigh
between 32 and 34 grams mathematically
we say this with the following notation
the probability of weighing a mouse
between 32 and 34 grams given the mean
of the distribution is 32 and the
standard deviation is 2.5 and all this
equals zero point two nine this is the
part of the equation we change if we are
interested in different Mouse weights
for example if we wanted to know the
probability of a mouse weighing more
than 34 grams we would change the bit on
the left side to reflect this the right
side which defines the shape and
location of the distribution stays the
same so when we talk about probabilities
we are talking about a distribution
that's described by the right side of
this equation and the area under the
curve that is described on the left side
using the same distribution we can
change the left side to get a new
probability BAM
now that we have probability worked out
let's talk about likelihood to talk
about likelihood you assume that you
have already weighed your mouse or mice
if you have weighed more than one
so here's our mouse it weighs 34 grams
the likelihood of weighing a 34 gram
Mouse is this point on the curve and
that value is 0.12 mathematically we say
this with the following notation the
likelihood of a distribution with mean
equals 32 and the standard deviation
equals 2.5 given we weighed a 34 gram
Mouse and all that equals 0.12
if we shifted the distribution over so
that the mean was 34 grams the new
likelihood would be zero point two one
so with likelihoods the measurements on
the right side are fixed and we modify
the shape and location of the
distribution with the left side double
bam
in summary probabilities are the areas
under a fixed distribution and
mathematically we have the probability
of data given a distribution likelihoods
are the y-axis values for fixed data
points with distributions that can be
moved mathematically this is written as
the likelihood of a distribution given
data if you want to see the actual
equations check out the stat quest that
derives the maximum likelihood estimator
for the exponential distribution hooray
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arm