 Hello, I am Deepika and I am going to help you to solve the following question. The question says, determine probability of E upon F. A die is thrown three times. E is the event four appears on the third toss and F is the event six and five appears respectively on first two tosses. Now we know that if E and F are two events associated with the same sample space of a random experiment, the conditional probability of the event E given that F has occurred that is probability of E upon F is equal to probability of E intersection F upon probability of F provided probability of F is not equal to zero. So this is a key idea behind our question. We will take the help of this key idea to solve the above question. So let's start the solution. Given question, a die is thrown three times. So the sample space of the given experiment consists of 216 outcomes. In the form of triplets yi zi where xi is equal to 1, 2, 3, 4, 5, 6 and yi is also equal to 1, 2, 6 and zi is equal to 1, 2, 6. Now we are given E is the event four appears on the third toss. So E is the set consists of elements 1, 1, 4, 1, 2, 4, 1, 3, 4, 1, 4, 4 till 1, 6, 4. Again 2, 1, 4, 2, 2, 4 till 2, 6, 4. Similarly 3, 1, 4, 3, 2, 4 till 3, 6, 4. Again 4, 1, 4, 4, 2, 4 till 4, 6, 4 and 5, 1, 4, 5, 2, 4 till 5, 6, 4. Similarly 6, 1, 4, 6, 2, 4 till 6, 6, 4. The set consists of elements which have 4 on the third throw. We are given F is the event that 6 and 5 appears respectively on first two tosses. So F is the set consists of elements which have 6 on the first and 5 on the second throw. So the elements of F are 6, 5, 1, 6, 5, 2, 6, 5, 3, 6, 5, 4, 6, 5, 5, 6, 5, 6. Clearly we observe that the set E intersection F consists of only one triplet which is 6, 5, 4. The probability of F is equal to 6 over 216 and probability of E intersection F is equal to 1 over 216. So by our key idea we have probability of E upon F is equal to probability of E intersection F over probability of F. Now probability of E intersection F is 1 over 216 and probability of F is 6 over 216. So probability of E upon F is equal to 1 over 216 into 216 over 6 and this is equal to 1 over 6. Hence the probability of E upon F is equal to 1 over 6. So this completes our session. I hope the solution is clear to you. Bye and have a nice day.