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State Vector — Definition, Formula & Examples

A state vector is a list of probabilities that tells you how likely a system is to be in each of its possible states at a given moment. For example, it might give the chance that tomorrow is sunny and the chance that it is rainy.

For a Markov chain with kk states, the state vector at step nn is a row vector xn=[ p1    p2    ⋯    pk ]\mathbf{x}_n = [\,p_1 \;\; p_2 \;\; \cdots \;\; p_k\,], where pi=P(system is in state i at step n)p_i = P(\text{system is in state } i \text{ at step } n). Every entry satisfies pi≥0p_i \ge 0, and ∑i=1kpi=1\sum_{i=1}^{k} p_i = 1. A vector with these properties is also called a probability vector.

Key Formula

xn+1=xnPsoxn=x0P n\mathbf{x}_{n+1} = \mathbf{x}_n P \qquad\text{so}\qquad \mathbf{x}_n = \mathbf{x}_0 P^{\,n}
Where:
  • xn\mathbf{x}_n = State vector (row vector) at step n
  • x0\mathbf{x}_0 = Initial state vector
  • PP = Transition matrix; entry in row i, column j is the probability of moving from state i to state j, and each row sums to 1
  • nn = Number of steps (a nonnegative integer)

Worked Example

Problem: A town's weather is either Sunny (S) or Rainy (R). If today is sunny, tomorrow is sunny with probability 0.8. If today is rainy, tomorrow is sunny with probability 0.3. Today's state vector is x0=[ 0.5    0.5 ]\mathbf{x}_0 = [\,0.5 \;\; 0.5\,]. Find tomorrow's state vector x1\mathbf{x}_1.
Step 1: Build the transition matrix. Row 1 is "from Sunny" and row 2 is "from Rainy." Each row must sum to 1.
P=[0.80.20.30.7]P = \begin{bmatrix} 0.8 & 0.2 \\ 0.3 & 0.7 \end{bmatrix}
Step 2: Multiply the row vector by PP. Each entry of the result is the dot product of x0\mathbf{x}_0 with one column of PP.
x1=[ 0.5(0.8)+0.5(0.3)      0.5(0.2)+0.5(0.7) ]=[ 0.55    0.45 ]\mathbf{x}_1 = [\,0.5(0.8) + 0.5(0.3) \;\;\; 0.5(0.2) + 0.5(0.7)\,] = [\,0.55 \;\; 0.45\,]
Step 3: Check that the result is a valid state vector. Both entries are nonnegative, and they add to 1.
0.55+0.45=10.55 + 0.45 = 1
Answer: x1=[ 0.55    0.45 ]\mathbf{x}_1 = [\,0.55 \;\; 0.45\,], so tomorrow there is a 55% chance of sun and a 45% chance of rain.

Why It Matters

State vectors let you predict long-run behavior. Examples include the share of customers who stay with a brand, or how a population moves between cities. Markov chains are not part of the core AP Statistics exam, but they show up in finite math, linear algebra, and probability courses, and they underlie models such as Google's original PageRank.

Common Mistakes

Mistake: Confusing a state vector with a unit vector, and scaling it so its length p12+⋯+pk2\sqrt{p_1^2 + \cdots + p_k^2} equals 1.
Correction: A state vector's entries must add to 1. Its length does not have to be 1. For example, [ 0.5    0.5 ][\,0.5 \;\; 0.5\,] is a valid state vector, but its length is 0.5≈0.707\sqrt{0.5} \approx 0.707.

Related Terms