Binomial Distribution - Probability, Class 12, Math Video Lecture | Mathematics (Maths) Class 12 - JEE

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today we're going to talk about the
binomial distribution in the binomial
test and they're going to be clearly
explained usually when people talk about
the binomial distribution they talk
about flipping a coin a coin usually has
heads in at least one tail for example
you can use the binomial distribution to
find out the probability of getting six
heads in six tosses but who really cares
about flipping coins what folks really
want to know is whether or not people
like orange Fanta more than grape Fanta
which flavor reigned supreme or are they
both equally loved to answer this
question we can ask a bunch of people
which flavor they prefer if everybody
but one person said they liked orange
Fanta more than grape Fanta then it
would be pretty obvious what people
liked most
but what if four people say they like
orange Fanta and three people say they'd
like grape Fanta is that enough to be
confident than most people like orange
Fanta or could it be that people in
general don't have a preference and
these results are just due to random
chance and a small sample size
maybe if we surveyed another seven
people we might only get three people
who like orange Fanta and for people who
like grape Fanta
to get to the bottom of this mystery we
need to get a sense of what to expect if
there is no preference
then we determine if our survey results
fit those expectations if not we can
reject the idea that both Fantas are
loved equally
the binomial distribution will tell us
what to expect if there is no preference
to say the same thing using statistics
lingo we will use the binomial
distribution aka this nasty looking
thing to model what to expect when there
is no preference then we'll see how well
this model fits the data if the model is
a poor fit we will reject the idea that
both flavors are loved equally
so let's start with a super simple
example and assume that I asked three
people if they liked orange Fanta more
than grape Fanta
the first person we asked said they
preferred orange Fanta the second person
we asked also said they preferred orange
Fanta
in the third person we asked said they
preferred grape Fanta if people really
didn't prefer one flavor over the other
then we will assume that there is a 50%
chance they will pick orange and a 50%
chance they will pick grape we can then
calculate the probability of the first
two people randomly choosing orange in
the third person randomly choosing grape
assuming that there is no real
preference the probability of the first
person preferring orange Fanta is 0.5 in
the probability of the first two people
preferring orange Fanta is 0.5 times 0.5
which equals 0.25 and the probability of
the first two people preferring orange
Fanta
and the third person performing grape is
0.5 times 0.5 times 0.5 which equals
0.125 note 0.125 is the probability of
the first two people saying they prefer
orange and the third person saying they
prefer grape it is not the probability
that any two out of three people would
prefer orange let me explain it could
have just as easily been that the first
person said they preferred grape
in this case the probability would still
be 0.125 but we'd multiply the numbers
together in a different order likewise
if the second person said they preferred
grape we just multiply the numbers
together in a different order so all
three of these combinations are equally
likely and this means that the
probability that any two out of three
people prefer orange Fanta is the sum of
the three possible orders so we just add
the three probabilities together in the
probability that any two out of three
people would randomly say they prefer
orange Fanta is 0.375 alternatively we
could have done the math using this
nasty-looking formula
X is the number of people who preferred
orange Fanta in this case X equals 2 n
is the total number of people we asked
in this case n equals 3 note n minus X
the total number of people we asked
minus the number of people who preferred
orange Fanta equals the number of people
who said they prefer grape Fanta
P is the probability that someone will
pick orange Fanta in this case P equals
0.5 note the probability that someone
might prefer grape Fanta is 1 minus P
together this says the probability of X
the number of people who say to prefer
orange Fanta given in the number of
people we asked and P the probability of
picking orange Fanta equals this nasty
looking thing ooh it's got factorials
don't freak out it looks fancy but it
just boils down to the number of
different ways two of three people could
say they prefer orange Fanta
when we did everything by hand we saw
that there were three ways for two of
three people to say they prefer orange
Fanta
and if we plug in N equals 3 and x
equals 2
and then just do the math
we get three
three ways that two out of three people
could prefer orange Fanta just like when
we did it by hand so this fancy thing is
really no big deal
the next part of the formula P to the X
corresponds to the probability that
orange Fanta was chosen two of the three
times in other words P to the X just
consolidates 0.5 times 0.5 into 0.5
squared
the last part of the equation
corresponds to the probability that
someone preferred grape Fanta remember
that one minus P is the probability that
someone prefers grape fine too
and n minus X is the number of people
that said they preferred grape Fanta
if we plug in N equals three x equals
two and P equals zero point five and
then do the math we get zero point five
so this term corresponds to the one
person who liked grape Fanta thus these
two parts of the equation correspond to
zero point five times zero point five
times zero point five
and the nasty part just multiplies it by
three
now we can put all the parts together
and plug in x equals to the number of
people that preferred orange Fanta in
equals three the number of people we
asked and P equals 0.5 the probability
someone would randomly pick orange Fanta
and we get the same probability that two
out of three people would randomly
prefer orange Fanta that we got when we
did everything by hand 0.375 in other
words the binomial distribution tells us
that the probability that two of three
people will prefer orange Fanta due to
random chance is 0.375
BAM
calculating the probability of three of
three people saying they prefer orange
Fanta by hand is pretty easy since there
is only one combination but we can just
as easily use the fancy formula by
plugging in x equals three and then we
just do the math this term equals one
since we are dividing three factorial by
three factorial and this term is also
equal one because anything raised to the
zero power equals one and then we just
keep doing the math
and this means that the probability of
three of three people randomly
preferring orange Fanta is 0.125
which is exactly what we got when we did
the calculations by hand
now that we've seen that we can
calculate probabilities with the
binomial distribution let's go back to
our original question if four people say
they'd like orange Fanta and three
people say they'd like grape Fanta can
we conclude that people in general
prefer orange Fanta
now we plug in x equals for the number
of people that preferred orange Fanta n
equals 7 the number of people we asked
and P equals 0.5 the probability someone
would randomly pick orange Fanta and
then just do the math
and we get zero point two seven three
the probability that four of seven
people would randomly prefer orange
Fanta double bam when you use a binomial
distribution to calculate a p-value it's
called a binomial test so what's the
p-value for four out of seven people
preferring orange Fanta the p-value is
the probability of the observed data
four of seven people prefer orange Fanta
plus the probabilities of all other
possibilities that are equally likely or
rare
this means we need to calculate these
probabilities these are the observed
results of our poll and these are rare
possibilities and we also need to
calculate the probabilities of these
combinations
these two possibilities for verses three
and three versus four are equally rare
if you don't believe me plug in the
numbers and see
the remaining possibilities are rare
in other words by including
possibilities when grape Fanta is
preferred equally or more often we are
calculating a two-sided p-value
if this is blowing your mind don't freak
out
just watch the stat quests on p-values
clearly explained and one and two sided
p-values the links are in the
description below we've already
calculated the probability that four out
of seven people prefer orange Fanta it's
zero point two seven three for this we
just set X to five and plug and chug and
we get zero point one six four then we
get zero point zero five five and then
we get zero point zero zero eight
adding the probabilities together gives
us 0.5 the probability that orange Fanta
is preferred now we just plug and chug
the numbers for when grape Fanta is
preferred
adding the probabilities together gives
us 0.5 the probability that orange Fanta
is not preferred the sum of the
probabilities of all combinations of
events that have an equal probability or
a rarer equals 0.5 plus 0.5 which equals
1 which means the p-value for 4 out of 7
people saying that prefer orange Fanta
is 1 which means that the model the
binomial distribution with P equals 0.5
ie orange Fanta and grape Fanta are both
equally loved is a good fit for the
observed data
thus we conclude that given the sample
size seven we cannot rule out the
possibility that both orange Fanta and
grape Fanta are equally loved
think about that the next time you watch
the World Series of baseball
triple bam
one last thing before we're done the
binomial distribution only works when
the probability that someone likes
orange Fanta does not change if someone
else already said they liked orange
Fanta in other words if we asked a bunch
of people if they liked orange Fanta and
they all say yes then that should not
affect the probability that the next
person also likes orange Fanta hooray
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