Feeling Sleepy

Discrete Mathematics Level 2

I am feeling sleepy but can't sleep. So I decide to roll a dice as many times as it takes until I get a 6 immediately after a 5.

Assuming each throw takes me one second and I will fall asleep as soon as I get a 6 immediately after a 5, what is the expected time (in seconds) in which I will fall asleep?

For example, if the results of the dice rolls are ${1, 6, 5, 6, 5, 2}$ , then I must stop at roll 4 and fall asleep immediately, which implies that the time taken would be 4 seconds.

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