The Monty Hall Problem

You are on a game show. There are three doors. Behind one is a car; behind the other two are goats. You pick door 1. The host — who always knows where the car is — opens one of the other doors to reveal a goat. He then offers you a choice: stick with door 1, or switch to the remaining closed door.

What do you do?

Most people say it doesn't matter. After all, two doors remain and one hides the car, so the odds must be 50-50. This intuition is wrong — and it was wrong even for many professional mathematicians. In 1990, when columnist Marilyn vos Savant published the correct answer in Parade magazine, she received nearly 10,000 letters disagreeing with her, including from PhDs. She was right. Switching wins 2/3 of the time.

The Monty Hall problem — named after the host of the US game show Let's Make a Deal — is one of the most famous probability puzzles ever devised. Its answer is proven, not controversial. Understanding why it's true is a lesson in conditional probability that reshapes how you think about information.

Play the game yourself, or run thousands of automated rounds and watch the win rates converge. The simulation runs the exact Monty Hall setup: random car placement, host always reveals a goat from the unchosen doors, you either switch or stay.

<p class="hint">{{hint}}</p>
<div class="doors" id="doors"></div>
<div class="msg" id="msg">{{choose_door}}</div>
<div class="btns" id="btns">
  <button id="switchBtn" type="button" disabled>{{btn_switch}}</button>
  <button id="stayBtn" type="button" disabled>{{btn_stay}}</button>
</div>
<div class="divider"></div>
<div class="stats">
  <div class="stat-block">
    <div class="bar-label">{{label_switching}}</div>
    <div class="bar-wrap"><div class="bar bar-switch" id="barSwitch"></div></div>
    <div class="stat-nums" id="numsSwitch">{{initial_stats}}</div>
  </div>
  <div class="stat-block">
    <div class="bar-label">{{label_staying}}</div>
    <div class="bar-wrap"><div class="bar bar-stay" id="barStay"></div></div>
    <div class="stat-nums" id="numsStay">{{initial_stats}}</div>
  </div>
</div>
<div class="btns auto-btns">
  <button id="auto100" type="button" class="ghost">{{btn_run100}}</button>
  <button id="auto1000" type="button" class="ghost">{{btn_run1000}}</button>
  <button id="resetAll" type="button" class="ghost danger">{{btn_reset}}</button>
</div>

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body { font-family: system-ui, sans-serif; color: #222; margin: 0; }
.hint { font-size: .9rem; color: #444; margin: 0 0 .8rem; line-height: 1.45; }
.doors { display: flex; gap: 12px; justify-content: center; margin: .5rem 0 .7rem; }
.door { width: 74px; height: 110px; border-radius: 10px; border: 2px solid #adb1b8;
        background: #c9ccd1; display: flex; flex-direction: column; align-items: center;
        justify-content: flex-end; padding-bottom: 10px; font-size: 1.1rem; font-weight: 700;
        cursor: pointer; user-select: none; transition: all .15s; position: relative; }
.door:hover:not([disabled]) { background: #bcc0c6; }
.door[disabled] { cursor: default; }
.door .emoji { font-size: 2.2rem; position: absolute; top: 18px; opacity: 0; transition: opacity .3s; }
.door.open .emoji { opacity: 1; }
.door.chosen { border-color: #1d3557; background: #dbe8f5; }
.door.goat-reveal { border-color: #e8a838; background: #fdf3dc; cursor: default; }
.door.win { border-color: #0a7d33; background: #d4f0e0; }
.door.lose { border-color: #c92f3c; background: #fde8ea; }
.door-num { font-size: .85rem; color: #555; }
.msg { font-size: 1rem; font-weight: 600; min-height: 1.4em; margin: .3rem 0 .5rem; text-align: center; }
.msg.ok { color: #0a7d33; }
.msg.bad { color: #c92f3c; }
.msg.info { color: #1d3557; }
.btns { display: flex; gap: .5rem; flex-wrap: wrap; justify-content: center; }
button { font: 600 14px system-ui, sans-serif; padding: .45rem .9rem;
         border: 1px solid #1d3557; background: #1d3557; color: #fff;
         border-radius: 8px; cursor: pointer; }
button.ghost { background: #fff; color: #1d3557; }
button.danger { border-color: #c92f3c; color: #c92f3c; }
button:disabled { opacity: .4; cursor: default; }
.divider { border-top: 1px solid #e0e4e8; margin: .9rem 0; }
.stats { display: flex; flex-direction: column; gap: .5rem; margin-bottom: .7rem; }
.stat-block { display: flex; align-items: center; gap: .6rem; }
.bar-label { width: 62px; font-size: .85rem; font-weight: 600; color: #333; text-align: right; }
.bar-wrap { flex: 1; background: #e8eef3; border-radius: 6px; height: 22px; overflow: hidden; }
.bar { height: 100%; border-radius: 6px; transition: width .4s ease; width: 0%; }
.bar-switch { background: #1d6d3a; }
.bar-stay { background: #c92f3c; }
.stat-nums { font-size: .8rem; color: #555; width: 130px; }
.auto-btns { margin-top: .3rem; }

// Code not found

After a few hundred rounds the pattern is unmistakable: staying wins roughly 1 in 3 times; switching wins roughly 2 in 3. The more rounds you run, the closer the bars get to those exact fractions. This is not a trick of the simulation — it is the mathematical truth playing out in real time.

The result is proven — not a simulation artifact. Here is why, presented three ways:

The case-by-case argument

Label the doors 1, 2, 3. You pick door 1. Each of the three cases is equally likely (probability 1/3):

1. Car is behind door 1 → host opens 2 or 3 → switching loses.
2. Car is behind door 2 → host must open door 3 → switching wins.
3. Car is behind door 3 → host must open door 2 → switching wins.

Two out of three cases mean switching wins. That's 2/3 — exact, not approximate.

The information argument

When you first picked, you had a 1/3 chance of being right. The host's reveal carries new information: he never opens the car door, so he is forced to avoid it. That constraint transfers the 2/3 probability that the car is not behind your door onto the single remaining unchosen door.

Bayes' theorem

Let $H_{3}$ = "host opens door 3", $C_{1}$ = "car is behind door 1". After the host opens door 3:

P($C_{1}$ | $H_{3}$) = P($H_{3}$ | $C_{1}$) · P($C_{1}$) / P($H_{3}$) = (1/2 · 1/3) / (1/2) = 1/3

The other door therefore holds the car with probability 2/3.

The key is that the host is not picking randomly — he is constrained by knowledge. A randomly-opening host who happened to reveal a goat would give you 50-50. Knowledge changes probability.

See how Bayesian inference generalizes this logic to any belief-updating scenario, or read why P vs NP asks whether finding answers is as easy as checking them.

The Monty Hall problem is a gateway to conditional probability — the probability of an event given that something else is already known. That idea appears everywhere:

• Medical testing: a positive test result does not mean you have the disease. The probability depends on the prevalence of the disease (the prior) and the false-positive rate. Ignoring the prior is called the base-rate fallacy, and it has led to real policy mistakes.
• Legal reasoning: the prosecutor's fallacy confuses P(evidence | innocent) with P(innocent | evidence). The Monty Hall intuition — that what the host knows and does changes probabilities — is exactly what juries sometimes fail to grasp.
• Spam filters: a Naive Bayes classifier updates the probability that an email is spam given each word it sees. The update rule is Bayes' theorem applied thousands of times per second.
• Machine learning: every probabilistic model — from Gaussian mixture models to large language models — is doing structured belief-updating. The intuition starts here.
• Decision theory: once you know that information changes probabilities, you can ask: how much is a piece of information worth? That question drives optimal experiment design, A/B testing and clinical trial methodology.

Master the Monty Hall problem and you have the key insight of Bayesian inference: new information reshapes old beliefs, often in ways that defy gut feeling.

The Monty Hall problem has a clean, proven answer: always switch. You win 2/3 of the time. The counter-intuitive feeling that it should be 50-50 comes from ignoring what the host knows — and that knowledge is the whole puzzle.

The deeper lesson is that probability is about information, not just counting outcomes. When an agent with knowledge acts on that knowledge, the probabilities seen by an observer shift. The Monty Hall host is a tiny, friendly version of the same mechanism that powers Bayesian classifiers, medical diagnostics and scientific inference.

So if you ever find yourself on a game show with three doors, you know what to do. And if you ever find yourself updating a belief in the light of new evidence, you now know the mathematics behind why that update might surprise you.