A box contains six red marbles,
four green marbles, and 10 black marbles.
Eight marbles are randomly selected with replacement.
Number one, what is the probability of getting four red,
two green, and two black marbles?
Number two, what is the probability of getting three red,
one green, and four black marbles?
These are examples of multinomial probabilities.
These resemble binomial probabilities,
except remember, with binomial probabilities,
there are only two possible outcomes,
success or not success.
Here, we have three possible outcomes,
depending on the color of the marble.
Because there are a total of 20 marbles,
six of which are red,
four of which are green, and 10 of which are black,
the probability of selecting one red marble is 3/10.
The probability of selecting one in green marble
is equal to 1/5.
And the probability of selecting one black marble
is equal to 1/2.
And again, we are selecting with replacement.
So for a multinomial probability,
there are N independent trials,
in our case because we are selecting eight marbles,
there are eight independent trials.
Each trial results in one of the K,
mutually exclusive outcomes,
which in our case should be the color.
On any single trial,
these K outcomes occur with the probabilities,
P sub one through P sub K,
which we just gave in the table.
The sum of the probabilities is equal to one
and the random variable X sub I,
represents the number of occurrences of I.
And we have the formula
for a multinomial probability at the bottom.
Let's go back to our examples.
Again, number one is,
what is the probability of getting four red,
two green, and two black marbles?
We can represent this as the probability,
that X sub one is equal to four.
This represents getting four red marbles.
Comma X sub two equals two.
This represents getting two green marbles.
Comma X sub three equals two.
This represents getting two black marbles.
And this is equal to,
we have N factorial divided by X sub one factorial,
through X sub K factorial.
So because we are selecting eight marbles with replacement,
we have eight factorial in the numerator,
and then because we want,
and because we want four red,
two green, and two black marbles,
we divide by the product of four factorial,
two factorial, and two factorial.
This gives us a number of ways of getting four red,
two green, and two black marbles.
Now we multiply this by,
the probability of getting a red,
raised to the fourth power,
which would be 3/10 to the fourth,
times the probability of getting a green marble squared,
which would be 1/5 squared.
And then times the probability of getting a black marble,
also squared, which is 1/2 squared.
So if we pause here for just a moment,
we have eight factorial here
because we are selecting eight marbles with replacement.
And then because we want four red marbles,
we have a four factorial here
and an exponent of four here.
Because we want two green marbles,
we have a factorial of two in the denominator here
and an exponent of two here.
And because we want two black marbles,
we have a two factorial in the denominator here
and an exponent of two here.
And notice how these probabilities here,
are the probability of getting a red,
the probability of getting a green,
and the probability of getting a black.
Let's go and evaluate this on the calculator.
Notice on the right, we have a decimal.
If we want the exact value as a fraction,
we simply click convert to fraction.
I'm going to go ahead and record the fraction,
as well as the decimal.
Which as a percentage,
would be 3.402%.
To convert a decimal to a percent,
we multiply by 100, add a percent sign.
Let's take a look at the second example.
We want the probability of getting three red,
one green, and four black,
which we can represent as the probability
that X sub one is equal to three,
X sub two is equal to one,
and X sub three is equal to four,
which is equal to,
we begin with eight factorial,
because we are drawing or selecting eight marbles.
And then because we want three red,
one green, and four black marbles,
we divide by the product of three factorial,
one factorial, and four factorial.
And then we multiply it by the probability,
of a red marble raised to the third power,
which is 3/10 to the third.
And then times the probability,
of a green marble raised to the first,
which is 1/5 to the first,
and then times the probability
of selecting a black marble raised to the fourth,
which is 1/2 raised to the fourth.
And now we go back to the calculator.
As a decimal, we have 0.0945, or as a fraction,
we have 189/2000.
Which as a percentage is 9.45%.
I hope you found this helpful.