Question Video: Finding the Probability of a Discrete Random Variable for a Given Value Mathematics

Let 𝑋 denote a discrete random variable which can take the values 2, 6, 7, and 8. Given that P(𝑋 = 2) = P(𝑋 = 6) = 3/22, and P(𝑋 = 7) = 4/11, find P(𝑋 = 8). Give your answer as a fraction.

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Video Transcript

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

Let’s begin by representing the information we’ve been given in a slightly different format. We can use a table to display this probability distribution function. In the top row, we’ll have the four values in the range of this discrete random variable, which are two, six, seven, and eight. In the second row, we’ll fill in the probabilities we’ve been given: three over 22 for both two and six and four over 11 for seven. We’re missing one of the probabilities: the probability that 𝑋 equals eight. And this is the value we’re 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 we’ve 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 𝑋 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 𝑋 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 𝑋 is equal to eight is four elevenths.

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