In problem for we cast the die to independent times and we let X. P. is the absolute value of the difference of the two resulting values. For example, if the two times gives one and four the X near equals three. We want to find the probability mass function of X here. It is not necessary to find a formula for the probability mass function. Then we can just get all possible values of X and find its probability. Then define the probability mass function. X can take Values ranges from 0 to 5 from zero. Can take 01, 2, 3, 4, 5. Because the maximum difference comes when the two numbers is one and six or six and one, then we have only two possibilities to have five and four will be for numbers for possibilities to have one and five or five and one of course two and 6 or 600. And when we get to three, These possibilities will be three. Not to blow by two will be six And for two will be former to blow by two is it? One will be five multiplied by two. Well beaten and zero. We have only six beers. We're not being multiplied by two Because six and 6 is the same as six answer. Then it's only six or 12 divided by two. These are the number of possibilities. And to get the probability We add by the submission and submission should be six multiplied by six 36. Six plus 10 plus eight plus six plus four plus 2. 36. Yes. Then It's one divided by six. Then by the bus at six It's five divided by 18. Then divided by 36. He stood divided by nine. Six and a quarter percent six. Of course one divided by six. four divided by 36 Is one divided by nine. and finally two divided by 36 is one divided by 18, and to find the probability mass function V. Of X. It equals one divided by sex, when X equality, five divided by 18. When X equals one, two divided by nine, and X equals two, one divided by six. x equals three. one, divided by nine. X equals four, wonderful by 18 When x equals five, and this is the final answer of our problem.