A probabilistic language.

How it works

Explore what could happen

Give Probl the choices and their chances. It follows the possible outcomes, combines paths that lead to the same situation, and answers the questions you ask.

1. 1 Describe the possibilities

   In this model, a 30% chance of rain leaves a 70% chance of sun. Write if 30% and Probl follows both possibilities, keeping track of how likely each one is.

   let weather = if 30% { "rain" } else { "sun" }report weather

   "rain"30%

   "sun"70%

2. 2 Combine paths that meet

   Flip a coin to step left or right. Left then right brings you back to the start, just like right then left. Probl combines those two paths: together, they give a 50% chance of being back at zero.

   When only your current position matters, Probl can skip keeping every route separately. That makes many models much smaller.

   var pos = 0repeat 3 { pos += if 50% { 1 } else { -1 } }report pos

3. 3 Ask a question

   Use report to ask what you want to know: the chance of rain, the odds of winning, or how long you might wait. Probl brings the possibilities together in one result.

   enumeratedweather    sun 70.00% · rain 30.00%

   sun

   rain

# Risk: 10 attacking armies against 8 defenders, fought to the end.@mode enumeratevar attackers = 10var defenders = 8while attackers > 1 and defenders > 0 {  # The attacker rolls up to three dice but must leave one army at home;  # the defender rolls up to two. roll(n, d6) gives the dice sorted high to low.  let attack ~ roll(min(3, attackers - 1), d6)  let defend ~ roll(min(2, defenders), d6)  # Highest against highest, second against second. Ties go to the defender.  for i in 0..<min(attack.len(), defend.len()) {    if attack[i] > defend[i] { defenders -= 1 } else { attackers -= 1 }  }}# A full round has 56 × 21 distinct dice outcomes. But once it's over, the# dice are never read again, so the only state left is the two army counts:# the worlds merge back into a handful after every round.report defenders == 0 as "attacker conquers"report attackers - 1 as "armies left to move in"

enumeratedattacker conquers         64.64%armies left to move in    mean 3.05 · sd 2.78 · 5% 0 · median 3 · 95% 8 · 0 █▁▂▃▃▃▃▂▂▁ 9

Keep rolling until you win or lose

In craps, your first roll can win, lose, or set a target. If it sets a target, keep rolling: hit that total again to win, or roll a seven to lose. Probl works out the overall chance of winning: 49.29%.

# Craps, pass line bet: what's the chance of winning?let come_out ~ 2d6               # one world per totalvar win = falseif come_out in [7, 11] {  win = true} else if come_out not in [2, 3, 12] {  loop {                         # keep rolling for the point    let r ~ 2d6    if r == come_out { win = true; break }    if r == 7 { break }  }}report win

enumeratedwin    49.29%

Forecasting

See a range of possible futures

How might revenue grow after a launch? Let growth and customer losses vary, run the model many times, and see both the likely outcomes and the wider range of possibilities.

sample · 50,000 runs · seed 11MRR ≥ $100k after 18 months    33.3% ± 0.2%

let price = 49                 # $ per customer per monthlet churn ~ 3% to 7%           # share of customers who leave each month: uncertain, but fixedvar market ~ one_of([Boom: 20%, Steady: 60%, Slump: 20%])var signups ~ 60 to 150        # new customers per month at launchvar customers = 0for month in 1..18 {  let new ~ poisson(signups)  let lost ~ binomial(customers, churn)  customers += new - lost  report customers * price by month as "MRR ($)"  market = next_month(market)  let growth ~ match market {  # month-on-month growth in sign-ups    Boom   => 8% to 20%    Steady => 1% to 6%    Slump  => normal_range(-8%, 0%)   # growth can be negative: say which shape you mean  }  signups *= 1 + growth}report customers * price >= 100_000 as "MRR ≥ $100k after 18 months"

Beyond dice and coins

Model waiting times and uncertain rates

An arrival time could be anywhere within half an hour. A success rate might be somewhere between zero and one. Probl can work with these ranges too, and calculate results directly for supported models.

let arrival ~ uniform(0, 30)observe arrival > 10let wait = max(arrival - 15, 0)report wait as "wait (minutes)"report wait > 10 as "waiting more than 10 minutes"

enumerated · evidence 66.67%wait (minutes)                  mean 5.62 · sd 4.96 · 5% 0.00 · median 5.00 · 95% 14.00waiting more than 10 minutes    25.00%

let rate ~ beta(2, 2)observe 3 from binomial(5, rate)report ratereport rate > 50%

enumerated · evidence 21.43%rate          mean 0.56 · sd 0.16 · 5% 0.29 · median 0.56 · 95% 0.81rate > 50%    63.67%

Interpreting results

Understand your results

enumerated · evidence 8.87%
sample · 50,000 runs · seed 11

let storm = bernoulli(30%)report storm and storm as "two independent storms"let happened ~ stormreport happened and happened as "the same storm twice"

enumeratedtwo independent storms    9.00%the same storm twice      30.00%

Get started

Run Probl

cargo install probl-cli --locked

let total ~ 2d6report total >= 10 as "Rolling 10 or more"

probl run first-roll.probl

Examples

22 example models

Examples cover games, forecasting, calendars, decisions, monitoring and risk. Each includes assumptions and expected results, and opens in the playground.

• 01 A tour of Probl Every core idea on one page
• 02Game Craps, pass line bet Draws, a loop with no fixed end, fractions
• 03Game RPG duel Records, functions, distributions as fields, simulate, a balancing sweep
• 04Game Risk Dice pools; merging collapses a huge tree
• 05Game Snakes and ladders Maps, long games, independent copies of a distribution
• 06Game Blackjack Cards without replacement, tables with report … by
• 07Forecasting SaaS launch a to b estimates, regime switching, a fan chart over months
• 08Forecasting Sign-ups observe … from, learning a rate, then forecasting with it
• 09Forecasting An illustrative forecast of Probl's original plan, not a current release schedule Risks and dates: an illustrative forecast of the original plan, not a release schedule
• 10Physics Quantum key distribution (BB84) A classical BB84 intercept–resend model; measurement probabilities and Bayes' rule, without complex amplitudes
• 11Forecasting Delivery forecast Delivery uncertainty from today, weekends, explicit holidays and deadline probability
• 12Forecasting Invoice cash flow Recurring invoices anchored to a calendar day, payment delays and monthly cash-flow buckets
• 13Forecasting Contract renewal Leap-day renewals, notice periods and uncertain response dates
• 14Decisions Stock decisions Expected profit, choosing before uncertainty resolves, perfect information and regret
• 15Operations Staffing a help desk Continuous arrivals and service times, shared scenarios for staffing comparisons
• 16Model checking Model checking Replicate a dataset from a fitted rate and compare variation between days
• 17Monitoring Sensor tracking Noisy observations of a hidden state, incremental exact filtering and next-state prediction
• 18Risk Shared hazards Common hazards, equal marginal risks, different joint tails and reserve shortfalls
• 19Arrival times Analytic continuous outcomes — arrival times without sampling Analytic continuous reports, shared draws and threshold evidence
• 20Game SET Monte Carlo simulation, batch draws without replacement, combinations and short-circuit search
• 21Forecasting Retirement A yearly balance with an absorbing zero, depletion risk by year and a constant-rate baseline
• 22Forecasting Correlated returns A joint normal vector, covariance, portfolio returns and loss probability