Final answer:
Among the scenarios, (b) the number of sixes in 'n' rolls and (c) the time since the most recent six are Markov chains. The largest number rolled up to the nth roll is not a Markov chain, nor is the time until the next six because it depends on future rolls. Option b, c.
Step-by-step explanation:
The concept we are focusing on is Markov chains, which in this context relate to the outcomes of rolling a die.
A Markov chain is a stochastic process that satisfies the Markov property, meaning the probability of transitioning to any state depends only on the current state and not on the sequence of events that preceded it. Let's explore the scenarios provided:
The largest number Xn shown on the nth roll is not a Markov chain since the probability of the largest number does not depend solely on the current roll but rather on all previous rolls.
The number Nn of sixes in n rolls is a Markov chain. The state space is {0, 1, 2, ..., n}, where each state represents the number of sixes obtained so far.
The transition from state i to state i+1 occurs if a six is rolled, and the probability is 1/6; otherwise, the state remains the same, with probability 5/6. The transition matrix will be a (n+1) x (n+1) matrix where the diagonals are 5/6, and the entries just above the diagonal are 1/6.
At time r, the time Cr since the most recent six is a Markov chain. The state space is {0,1,2,...,r} because each state indicates the time passed since the last six rolled. The transition probability from any state to state 0 (rolling a six) is 1/6, and moving from state i to i+1 (not rolling a six) is 5/6.
At time r, the time Br until the next six is not a Markov chain, because it depends on the future outcome which is not known at the present state.
Example transition matrix for the scenario b with n=2:
[5/6 1/6 0]
[0 5/6 1/6]
[0 0 1]
So Option b,c.