### Video Transcript

Let ๐ denote a discrete random
variable which can take the values two, six, seven, and eight. Given that the probability ๐
equals two is equal to the probability ๐ equals six, which is three over 22, and
the probability ๐ equals seven is four elevenths, find the probability ๐ equals
eight. Give your answer as a fraction.

Letโs begin by representing the
information weโve been given in a slightly different format. We can use a table to display this
probability distribution function. In the top row, weโll have the four
values in the range of this discrete random variable, which are two, six, seven, and
eight. In the second row, weโll fill in
the probabilities weโve been given: three over 22 for both two and six and four over
11 for seven. Weโre missing one of the
probabilities: the probability that ๐ equals eight. And this is the value weโre asked
to find.

To do so, we need to recall that
the sum of all the probabilities in a probability distribution function must be
equal to one because the discrete random variable can only take values within its
range. We can therefore form an equation
and then substitute the three probabilities weโve been given in the question. By thinking of four over 11 as the
equivalent fraction eight over 22, we then have the equation 14 over 22 plus the
probability ๐ equals eight is equal to one. Subtracting 14 over 22 from each
side and then simplifying 14 over 22 to seven over 11, we have that the probability
๐ equals eight is equal to one minus seven over 11, which is four over 11.

So, by using the fact that the sum
of all the probabilities in a probability distribution function must be equal to
one, we found the missing probability. The probability that ๐ is equal to
eight is four elevenths.