Pick 5

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, not value. 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

You pick 1-2-3-4-5: all distinct, so ways = 5! = 120 - a 120-way ticket.

Box payout = $50,000 / 120 = $416.67. Box probability = 120/100,000 = 3/2,500. Straight probability = 1/100,000.

EV audit: Straight (1/100,000) x $50,000 = $0.50. Box (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. House edge = 50% on every row.

The draw comes 5-4-3-2-1. Straight misses. Box hits - the same five digits in reverse - paying $416.67 on the $1 wager. Pair (drawn 5-4 vs your 1-2), Front 4, and Back 4 all miss. A 0.12% shot landed; the ticket was still a 50-cent asset at purchase.

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 - rather than re-deriving each case from scratch - is exactly the abstraction step that separates a good interview answer from a 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.