Pick 5 Lottery Math Guide

How it works

You fill in a 5-digit ticket, one digit (0-9) per column, on a $1 wager, then answer questions generated from your own ticket before the draw - the ordering count, the implied Box payout, the Straight and Box probabilities, and the house-edge computation. The payout model was verified against a real state Pick 5 prize table.

Five digits are drawn uniformly at random and checked against every play type at once: Straight (exact order, 1 in 100,000), Box (same digits any order), Pair (first two digits in order, 1 in 100), and Front 4 / Back 4 - your first or last four digits in exact order at 1 in 10,000 each.

Ways = 5! divided by repeat factorials: five distinct digits are a 120-way; a single pair is a 60-way (5!/2!); a triple is a 20-way; and so on down to 1-way for five of a kind.

How scoring works

Straight pays $50,000 (10^5 / 2) at probability 1/100,000.

Box pays $50,000 / ways - $416.67 on a 120-way, $833.33 on a 60-way - at probability ways/100,000.

Pair pays $50 at probability 1/100. Front 4 and Back 4 each pay $5,000 (10^4 / 2) at probability 1/10,000.

Each payout is exactly half the fair payout for its odds, so EV per $1 is $0.50 on every play type - a uniform 50% house edge.

The jackpot scales; the edge does not

Straight EV: (1/100,000) x $50,000 = $0.50. Box on a 120-way: (120/100,000) x $416.67 = $0.50. Pair: (1/100) x $50 = $0.50. Front 4: (1/10,000) x $5,000 = $0.50. From Pick 3's $500 to Pick 5's $50,000 the jackpot grew 100x while its probability shrank 100x - the EV never moved off 50 cents.

This is the lab's cleanest demonstration that jackpot size is marketing. A fair Straight would pay $100,000; the game, like real lotteries, pays half.

Big factorials, same recipe

Ways = 5!/repeats: 1-2-3-4-5 gives 120; 1-1-2-3-4 gives 5!/2! = 60; 1-1-2-2-3 gives 5!/(2!2!) = 30; 1-1-1-2-3 gives 5!/3! = 20. Box probability is ways/100,000 and Box payout is $50,000/ways, so the two questions answer each other: once you have the way count, both numbers are one division away.

A 120-way Box at 120/100,000 = 0.12% is the most winnable non-Pair bet on the ticket, and it still returns 50 cents per dollar in expectation.

Front/Back 4 nests the whole family

A Front 4 bet is a Pick 4 Straight living inside your Pick 5 ticket: 1/10,000 for $5,000. Pick 5 therefore contains Pick 4's headline bet (Front/Back 4), the Pair from Pick 3, and its own Straight - a nested family of bets whose EV table is a constant. When one row of the table looks tempting, recompute it: (probability) x (payout) = $0.50, always.

A worked example

1. Setup: ticket 1-2-3-4-5 has five distinct digits, so its orderings are the full factorial.

   $$1\text{-}2\text{-}3\text{-}4\text{-}5: \quad \text{ways} = 5! = 120$$

2. The way count fixes the Box payout and the Box/Straight probabilities out of 100,000 possible draws.

   $$\text{Box payout} = \frac{\$50{,}000}{120} \approx \$416.67, \qquad P(\text{Box}) = \frac{120}{100{,}000} = \frac{3}{2{,}500}, \qquad P(\text{Straight}) = \frac{1}{100{,}000}$$

3. EV of Straight and Box both land on the same 50 cents on the dollar, despite the huge gap in jackpot size and odds.

   $$EV_{\text{Straight}} = \frac{1}{100{,}000} \times \$50{,}000 = \$0.50, \qquad EV_{\text{Box}} = \frac{120}{100{,}000} \times \$416.67 \approx \$0.50$$

4. EV of Pair and Front 4 confirm the same flat edge across the whole ticket.

   $$EV_{\text{Pair}} = \frac{1}{100} \times \$50 = \$0.50, \qquad EV_{\text{Front 4}} = \frac{1}{10{,}000} \times \$5{,}000 = \$0.50$$

5. The draw comes 5-4-3-2-1: wrong order for Straight, Pair, Front 4, and Back 4, but the same multiset as your ticket, so Box hits and pays out.

   $$\{5,4,3,2,1\} = \{1,2,3,4,5\} \ \Rightarrow\ \text{Box hit, pays } \$416.67$$

Common mistakes

• Chasing the $50,000 headline. At 1/100,000 it prices to the same $0.50 EV as every other bet in the lab's three lotteries.

• Forgetting repeat adjustments at n = 5: 1-1-2-2-3 is 30-way (5!/(2!2!)), not 60 or 120.

• Confusing Box with Front/Back. Box is any order over all five digits; Front 4 and Back 4 are strict in-order positional matches of four.

• Thinking distinct digits are 'smarter' picks. A 120-way ticket wins Box more often and pays proportionally less - EV is identical to a 1-way quint ticket.

• Comparing lotteries by jackpot. Pick 3, 4, and 5 differ only in variance; the house edge is a flat 50% across all of them.

Why interviews test this

Pick 5 completes the scaling argument interviewers often walk candidates through: as n grows, payoff and improbability scale together and expected value is invariant. Recognizing an invariant across a family of bets is the abstraction step that separates a good interview answer from a merely correct one.

It is also a lesson in variance preference: all three Pick games are 50%-edge bets, so choosing among them is purely a choice of payoff distribution. Interviewers use setups like this to ask utility and risk questions, such as why anyone buys the highest-variance version of a strictly losing bet.

Play Pick 5 on Outcry

Pick 5 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 5 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.