Let 𝑋 denote a discrete random
variable which can take the values negative two, two, four, and five. Given that the probability 𝑋
equals negative two is 0.15, the probability 𝑋 equals two is 0.43, and the
probability 𝑋 equals four is 0.25, find the probability 𝑋 is greater than two.
In this question, we’re not asked
to determine the probability that our random variable is equal to a particular
value, but rather the probability that it is greater than a given value. This may look a little confusing at
first, but the key to remember is that a discrete random variable can only take the
values in its range and no others. Let’s begin by representing
information in the question using a table. We have the values in the range of
our discrete random variable in the top row and the corresponding probabilities in
the second row.
Now, we do have a missing
probability, the probability that 𝑋 equals five, but we’ll worry about that later
if necessary. We want to find the probability
that 𝑋 is greater than two. Now remember that our discrete
random variable can only take the values in its range. For 𝑋 to be strictly greater than
two, this means that it must either take the value four or the value five. There are no other values it can
take. The probability 𝑋 is greater than
two then is equal to the sum of the probability 𝑋 equals four and the probability
𝑋 equals five.
Now remember, we don’t know the
probability 𝑋 equals five, although we can easily calculate it by remembering that
the sum of all probabilities in the distribution must be equal to one. Instead, a slightly easier way may
be to recall that the probability 𝑋 is greater than two will be equal to one minus
the probability 𝑋 is less than or equal to two. So, instead, we can just subtract
the probabilities for 𝑋 equaling negative two and 𝑋 equaling two from one, one
minus 0.15 and 0.43, which is equal to 0.42.