Probability, conditional probability, Bayes (lightly)

Probability — what 95% actually means

You hear it constantly. "70% chance of recession next year." "Market is pricing in a 90% probability of a rate cut." "There's a 5% chance this option ends in the money." The number sounds precise. What it's actually saying is much fuzzier than people think.

Probability is just a number between 0 and 1 (or, in friendly outfit, between 0% and 100%) that represents how confident we are that something happens. 0 means impossible. 1 means certain. Everything else is some shade of "maybe."

That's it. There's no quantum mystery. Probability is just a confidence number on a sliding scale.

Where probability gets useful: it makes "expected value" work

You're offered a coin flip. If heads, you win $20. If tails, you lose $10. Should you play?

The expected value tells you. Half the time you win $20: 0.5 × $20 = $10. Half the time you lose $10: 0.5 × (−$10) = −$5. Add them: $10 + (−$5) = $5. That's the expected value of one play.

Positive expected value → over many plays, you come out ahead. Now picture the probability and the payoff as two pieces of a single number. Expected value = probability × payoff, summed over all outcomes. That's the engine behind every "should I take this trade" question in finance.

Conditional probability — when one thing depends on another

Most real-world probabilities aren't standalone. They depend on other things.

"Probability the stock rises tomorrow" → not very informative. "Probability the stock rises tomorrow given the Fed cut rates today" → much more useful. The "given X" part is the conditioning. Written as P(stock up | Fed cut). The vertical bar reads as "given that."

This shows up everywhere in finance. Implied volatility is conditional on current option prices. Earnings expectations are conditional on guidance. Bond prices are conditional on rate expectations. Most of finance is conditional probability in disguise.

Bayes — updating beliefs as evidence comes in

This is the part that scares people, but the picture is dead simple. You start with a prior belief. New evidence arrives. You update.

Example: before NVIDIA's earnings, you think there's a 50/50 chance it'll beat expectations. Earnings come out: it beat by 30%. Now what's the probability the company is fundamentally healthy? Higher than before, obviously. Bayes' theorem is the math that says by how much.

The exact formula:

Don't memorize this. Recognize the shape: probability of A given B, in terms of probability of B given A, scaled by their priors. The takeaway is just: Bayes is how rational people update their beliefs when new information arrives. Markets do this constantly. So should you.

Why this matters for finance

Almost everything in markets is a probability statement, even when it doesn't sound like one:

• "Stock is undervalued" = "I think there's a high probability the price will rise."
• "Recession risk" = a conditional probability.
• Option prices = literally weighted by probability distributions.
• "Bayesian investor" = someone who explicitly updates as data arrives.

You don't need to compute anything by hand. You need to recognize when an investing claim is implicitly a probability claim, and to feel comfortable with the question "by how much should this evidence update my view?"

The takeaway

Probability is a confidence number from 0 to 1. Conditional probability is "given X." Expected value is probability times payoff, summed across outcomes. Bayes is the math of updating beliefs as new evidence arrives. All of these will show up in Tracks 8 (derivatives) and 9 (portfolio). For now, just lock in the picture: probability is just a number.