 Hi friends, I am Pooja and today we will discuss the following question. Find the probability distribution of the number of successes in two tosses of a die where the success is defined as number greater than 4. Let us begin with the solution now. Now let S be the sample space of the experiment of tossing two dies or you can say two tosses of a die. Then S is the set which consists of the following 36 outcomes 1, 1, 1, 2 and so on up till 1, 6, 2, 1, 2, 2 so on up till 2, 6, 3, 1, 3, 2 so on up to 3, 6, 3, 6, 4, 1, 4, 2 so on up to 4, 6, 5, 1, 5, 2 so on up to 5, 6, 6, 1, 6, 2 so on up to 6, 6. Let X denote the event that the number greater than 4 turns up. Then three cases arise. The first one being that both the numbers that is the first number which appears on the first toss of a die and the second number which appears on the second toss of a die are either less than or equal to 4. The second case being that out of the two numbers one number greater than 4 turns up and the third case being that both the numbers are greater than 4. Now since X denotes the event that the number greater than 4 turns up so going by the three cases we get X is equal to 0, 1, 2. Now when X is equal to 0 that is when both the numbers are either less than or equal to 4 then the possible outcomes are 1, 1, 1, 2, up till 1, 4, 2, 1, 2, 2, up till 2, 4, 3, 1, 3, 2, up till 3, 4, 4, 1, 4, 2, up till 4, 4. That is when X is equal to 0 then these are the 16 possible outcomes so we get probability of X is equal to 16 upon 36 because out of total 36 outcomes there are 16 outcomes in which both the numbers are either less than or equal to 4 and we get this is equal to 4 upon 9. When X is equal to 1 that is when one number greater than 4 turns up then the possible outcomes are 1, 5, 1, 6, 2, 5, 2, 6, 3, 5, 3, 6, 4, 5, 4, 6, 5, 1, 5, 2, up to 5, 4, 6, 1, 6, 2, up to 6, 4. That is these are the 16 possible outcomes in which one number is greater than 4 so we get probability of X is equal to 16 upon 36 because out of total 36 outcomes there are 16 possible outcomes in which one number greater than 4 turns up so we get probability of X is equal to 16 upon 36 and we get this is equal to 4 upon 9. Now when X is equal to 2 that is when both the numbers are greater than 4 then the possible outcomes are 5, 5, 5, 6, 6, 5, 6, 6. That is these are the only four possible outcomes in which both the numbers are greater than 4 so we get probability of X is equal to 4 upon 36 because out of total 36 outcomes there are only four outcomes in which both the numbers are greater than 4 and we get this is equal to 1 upon 9 so this is our probability distribution table. From this table we can clearly see that when X is equal to 0 probability of X is equal to 4 upon 9 when X is equal to 1 probability of X is equal to 4 upon 9 and when X is equal to 2 probability of X is equal to 1 upon 9 So this is our answer. Hope you have understood the solution. Bye and take care.