Please help with proba question. What is the probability that the game ends on an odd-numbered of roll?

3. A game of dice is played by repeatedly rolling a fair die (where outcomes {1, 2, 3, 4, 5, 6} are equally likely on each roll, and the result of any roll does not affect the result of any other roll) until the number 1 or the number 6 is rolled. What is the probability that the game ends on an odd-numbered of roll ? 4 MARKS Hint: here the sample space S is the collection of sample outcomes each of which describes when the game ends; for 75 2 1, define the elementary event E,- to be \"the game ends on roll 2'\". Let the event A be the event \"the game ends on an odd-numbered of roll\". Then we have the (countable) partitions m 00 S: UE,' and A: UE2+1 '22:]. 11:0 We need to find P(A). Suppose the game ends on roll five; we must have had four consecutive rolls with results that are NOT 1 or 6, and then one roll where the result IS 1 or 6. By the conditions of the experiment, all sequences of ve rolls are equally likely and there are 65 possible such sequences (we can get any of the results {1, 2, 3, 4, 5, 6} on each of the five rolls); of these sequences, only a subset meet the condition that the rst four rolls do not give 1 or 6, but the last one does give 1 or 6. To compute the probability of the game ending on roll five, you need to count the number of sequences in the subset