Pick 3 Lottery Math Guide

How it works

You fill in a 3-digit ticket, one digit (0-9) per column, on a $1 wager - the same mechanics as a real state Pick 3, and the payout table is modeled on a real state prize table. Before the draw you answer a short question set generated from your own ticket: how many orderings your digits have, what the Box payout should be, the Straight and Box probabilities, and the house-edge punchline.

Three digits are then drawn uniformly at random (each 0-9, independent). Your ticket is checked against every play type at once: Straight (exact order), Box (same digits, any order - only counted when it is not already a Straight), and Pair (your first two digits match the first two drawn, in order). A single draw can hit multiple play types.

The 'way' count is the number of distinct orderings of your digits: 3 distinct digits are a 6-way, a pair like 1-1-2 is a 3-way, and triples are 1-way. It determines the Box payout and probability. As the game's final question forces you to prove, it does not change the value of the bet.

How scoring works

Straight pays $500 (10^3 / 2) at probability 1/1000.

Box pays $500 divided by your way count - $83.33 on a 6-way, $166.67 on a 3-way - at probability ways/1000.

Pair pays $50 at probability 1/100.

Every payout in the table is exactly half the fair payout for its odds: EV per $1 is $0.50 on every play type, a 50% house edge with no exceptions.

Prove the 50% edge once, then stop looking for a good bet

Straight: EV = (1/1000) x $500 = $0.50 per $1. Box on a 6-way: EV = (6/1000) x $83.33 = $0.50. Box on a 3-way: (3/1000) x $166.67 = $0.50. Pair: (1/100) x $50 = $0.50. Fair payouts would be $1000, $166.67 ($1000/6), and $100 respectively; the game pays exactly half of fair everywhere.

This is the edge insight the last question asks for outright: house edge = 1 - EV per dollar = 0.5 for EACH bet. The way count and payout size just redistribute the same $0.50 of expected value across more or fewer winning combinations. No digit choice, play type, or 'due number' changes it.

Way counts are permutations with repeats

Ways = n! divided by the product of factorials of each repeated digit's count. For 3 digits: all distinct gives 3! = 6; one pair gives 3!/2! = 3; a triple gives 3!/3! = 1. The Box probability is ways/1000 because exactly that many of the 1000 equally likely draws contain your multiset of digits.

This is the interview-ready skill in the game: translating 'how many orders' into a factorial expression in your head, then immediately converting a count into a probability by dividing by the size of the sample space.

Multiple hits do not rescue the EV

One draw is checked against every play type, so a Straight hit also pays your Pair if the first two digits match. But each play type is a separate $0.50-EV component - stacking bets stacks identical 50% edges. The lottery structure has no interaction term that helps you.

A worked example

1. Setup: ticket 1-2-3 has three distinct digits, so count its orderings with a factorial.

   $$1\text{-}2\text{-}3: \quad \text{ways} = 3! = 6$$

2. The way count sets both the Box payout (jackpot split ways-many times) and the Box/Straight probabilities out of the 1,000 possible draws.

   $$\text{Box payout} = \frac{\$500}{6} \approx \$83.33, \qquad P(\text{Straight}) = \frac{1}{1000}, \qquad P(\text{Box}) = \frac{6}{1000} = \frac{3}{500}$$

3. EV of the Straight bet: probability times payout.

   $$EV_{\text{Straight}} = \frac{1}{1000} \times \$500 = \$0.50$$

4. EV of the Box bet comes out the same, since the higher probability is offset by the proportionally smaller payout.

   $$EV_{\text{Box}} = \frac{6}{1000} \times \$83.33 \approx \$0.50$$

5. EV of the Pair bet matches too, giving the flat house edge across every play type.

   $$EV_{\text{Pair}} = \frac{1}{100} \times \$50 = \$0.50, \qquad \text{house edge} = 1 - 0.50 = 50\%$$

6. The draw comes 2-1-3: wrong order for Straight and Pair, but the same multiset as your ticket, so Box hits and pays out.

   $$\{2,1,3\} = \{1,2,3\} \ \Rightarrow\ \text{Box hit, pays } \$83.33$$

Common mistakes

• Believing a 1-way (triple) Straight-style Box is a better bet because it pays $500 instead of $83.33. Its probability is 1/1000 instead of 6/1000 - the EV is the same $0.50.

• Computing Box probability as 1/1000. It is ways/1000; forgetting the permutation count is the most common miss in the question set.

• Forgetting to divide by repeats. 1-1-2 has 3!/2! = 3 orderings, not 6.

• Reading the Pair bet as a side hedge. It is the same 50% edge in a smaller package: (1/100) x $50 = $0.50.

• Playing 'due' digits. Draws are independent uniform digits; no history changes the 1/1000.

Why interviews test this

Lottery pricing is a standard interview warm-up because it packs three testable skills into one prompt: counting permutations with repeated elements, converting counts to probabilities, and comparing an offered payout to the fair payout implied by those odds. The 'every play type has the same edge' punchline is exactly the kind of invariant interviewers love - candidates who compute one EV and assert the pattern show structural thinking; candidates who grind all four show reliability; the best do both.

It is also the cleanest possible example of why EV is the right lens on any priced bet: a 50% house edge is invisible in any single draw and completely decisive over volume.

Play Pick 3 on Outcry

Pick 3 is free to play on Outcry, in the Quitters Never Lose section, and every run is timed and scored. Read the guide once, then play Pick 3 and check your score against the strategy above. The guide book has a written guide for every other game, and the practice hub and mock assessments put the same skills under a real screen's clock.