Core Statistical Model Interactive
Markov Chains
Model state transitions where the future depends only on the current state. Perfect for streaks, game flow, and user behavior modeling.
📊 The Markov Property
P(X_n+1 | X_n, X_n-1, ...) = P(X_n+1 | X_n)
The next state depends only on the current state, not on history. "Memoryless" property simplifies many real-world models.
- • Win/Lose streaks
- • Game momentum
- • User engagement states
Transition Matrix
| To: Win | To: Lose | |
|---|---|---|
| From: Win | 0.60 | 0.40 |
| From: Lose | 0.45 | 0.55 |
Transition Probabilities
0.3 0.9
0.3 0.9
Higher P(Win→Win) = stronger hot streaks
Higher P(Lose→Lose) = harder to break slumps
📊 Steady State
Long-run probabilities (infinite time):
52.9%
Win
47.1%
Lose
Simulated State Sequence
L
W
L
L
W
W
W
L
L
L
W
L
L
W
W
W
L
L
W
W
L
W
W
W
W
W
W
L
L
W
W
L
L
L
W
W
W
W
W
W
W
L
W
W
L
L
L
W
W
W
Observed Win Rate: 60.0% vs Steady State: 52.9%
Cumulative Wins Over Time
🏀 Sports Betting Applications
Hot Streaks
Model probability of continuing winning/losing streak
Injury Status
Healthy → Injured → Out transition probabilities
Game Flow
Leading → Close → Trailing state transitions
User Activity
Active → Churned → Reactivated states
R Code Equivalent
# Markov chain simulation
library(markovchain)
# Define transition matrix
tm <- matrix(c(0.6, 0.4, 0.44999999999999996, 0.55),
nrow = 2, byrow = TRUE)
rownames(tm) <- colnames(tm) <- c("Win", "Lose")
# Create Markov chain
mc <- new("markovchain", transitionMatrix = tm)
# Steady state
steady <- steadyStates(mc)
cat(sprintf("Steady state: Win=%.1f%%, Lose=%.1f%%\n",
steady[1] * 100, steady[2] * 100))
# Simulate
sim <- rmarkovchain(n = 50, mc, t0 = "Win")
table(sim) / length(sim)✅ Key Takeaways
- • Markov: next state depends only on current
- • Transition matrix captures all probabilities
- • Steady state = long-run distribution
- • Model streaks, game flow, user states
- • Estimate streak/slump probabilities
- • Foundation for more complex models