In a two state Markov model, 1 - P(Dt|Wt-1) = ?

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BlackTom AI · Accepted Answer

The question concerns a two-state Markov model with states D and W (for example, D for default/drought and W for wet or another state). Since there are only two possible states, knowing one state at time t-1 (Wt-1) constrains the possible state at time t to either D or W, and these two possibilities are mutually exclusive and cover all outcomes.

Optionally, we can use the complement rule: the probability of D at time t given Wt-1 plus the probability of not-D at time t given Wt-1 must add up to 1, because exactly one of the two states must occur at time t. If we interpret not-D at time t as W at time t (since the two states are D and W and there are no other states), then:
1 - P(Dt | Wt-1) = P(not-Dt | Wt-1) = P(Wt | Wt-1).

Thus, in a two-state model where the two states are exhaustive and mutually exclusive, subtracting the probability of D from 1 yields the probability of W at time t given the previous state Wt-1, assuming the conditioning is appropriate and the model allows W as the complementary state.

If one wonders about edge cases, this reasoning relies on there being exactly two states (D and W) with no additional states or absorbing conditions. If more states were present, 1 - P(Dt | Wt-1) would not necessarily equal P(Wt | Wt-1).

Note: The prompt’s answer list is empty, so this reasoning addresses the relationship conceptually and shows why the expression 1 - P(Dt|Wt-1) corresponds to P(Wt|Wt-1) under a two-state, binary-state framework.

ENEN20002_2025_SM1 Quiz on Lecture 6 material

In a two state Markov model, 1 - P(Dt|Wt-1) = ?

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