Every financial decision is a probability calculation, whether you realize it or not. Should you buy insurance? Depends on the probability and cost of the event. Should you diversify investments? Depends on the probability of any single investment failing. Should you take the lottery? Depends on the probability-weighted value of the ticket. Here is how to think and calculate more clearly.

Basic Probability

Probability = Favorable Outcomes / Total Possible Outcomes

A standard die: probability of rolling a 4 = 1/6 = 0.1667 = 16.67%.

Compound Events

P(A AND B) — independent events: P(A) x P(B)

Probability of flipping heads twice in a row: 0.5 x 0.5 = 0.25 = 25%.

P(A OR B): P(A) + P(B) – P(A AND B)

Probability of rolling a 1 or a 2 on a die: 1/6 + 1/6 – 0 = 2/6 = 33.3%.

Expected Value in Financial Decisions

Expected Value = Sum of (Outcome x Probability of That Outcome)

A lottery ticket costs $2. There is a 1 in 300,000,000 chance of winning $500,000,000 and various smaller prize probabilities. EV = ($500M x 1/300M) + (smaller prizes) – $2 ticket cost. Most lottery tickets have an EV of $0.50-0.90 per $1 spent — a guaranteed long-term loss.

Insurance, by contrast, can be positive EV when the potential loss is catastrophic. A $1,500/year homeowner’s policy on a $400,000 home makes sense even if the probability of a major claim is only 0.5%/year — because $400,000 x 0.005 = $2,000 expected annual loss, and $1,500 to protect against that is rational.

Combinations in Finance

How many ways can 3 stocks be selected from a portfolio of 10? C(10,3) = 10! / (3! x 7!) = 120 combinations. This matters for understanding portfolio diversification — the number of unique pairwise correlations in a 10-stock portfolio is C(10,2) = 45, each of which affects overall portfolio volatility.

The Gambler’s Fallacy

After 10 consecutive coin flips landing heads, the probability of the 11th flip landing tails is still 50%. Each flip is independent. Past outcomes do not influence future probabilities for independent events. This error leads investors to believe a stock “must bounce back” after consecutive losses — when each day’s return is largely independent of prior days.