Pick 4

How it works

You fill in a 4-digit ticket, one digit (0-9) per column, on a $1 wager. Before the draw you answer questions generated from your own ticket - orderings, the implied Box payout, Straight and Box probabilities, and the house-edge computation.

Four digits are drawn uniformly at random and checked against every play type at once: Straight (exact order, 1 in 10,000), Box (same digits any order, ways in 10,000), Pair (first two digits in order, 1 in 100), and - new at 4 digits - Front 3 and Back 3, which match your first or last three digits in exact order at 1 in 1,000 each.

Ways = 4! divided by repeat factorials: 4 distinct digits are a 24-way, one pair is a 12-way (4!/2! = 12), two pairs are a 6-way, a triple is a 4-way, and quads are 1-way.

How scoring works

Straight pays $5,000 (10^4 / 2) at probability 1/10,000.

Box pays $5,000 / ways - $208.33 on a 24-way, $416.67 on a 12-way - at probability ways/10,000.

Pair pays $50 at probability 1/100. Front 3 and Back 3 each pay $500 (10^3 / 2) at probability 1/1,000.

Every payout is half of fair for its odds: EV per $1 is $0.50 across the board - a uniform 50% house edge.

Run the EV table - it is flat at $0.50

Straight: (1/10,000) x $5,000 = $0.50. Box, 24-way: (24/10,000) x $208.33 = $0.50. Box, 12-way: (12/10,000) x $416.67 = $0.50. Pair: (1/100) x $50 = $0.50. Front 3 or Back 3: (1/1,000) x $500 = $0.50. Fair payouts would be exactly double each of these - $10,000, $416.67, $833.33, $100, $1,000.

The jump from Pick 3 to Pick 4 multiplies the Straight jackpot by 10 and divides its probability by 10. The product - the only thing EV cares about - does not move.

Master the repeat-adjusted way count

Ways = 4! / product of repeat factorials. 1-2-3-4: 24. 1-1-2-3: 4!/2! = 12. 1-1-2-2: 4!/(2!2!) = 6. 1-1-1-2: 4!/3! = 4. 1-1-1-1: 1. The Box question set hinges on producing this number quickly, then reading Box probability as ways/10,000 and Box payout as $5,000/ways.

Notice the see-saw: more ways means a higher hit probability and a proportionally smaller payout. The product is pinned at $0.50 by construction - which is the final question's answer.

Front/Back 3 is Pick 3 hiding inside Pick 4

A Front 3 bet on a 4-digit ticket is exactly a Pick 3 Straight on your first three digits: 1/1,000 for $500. Same for Back 3 on the last three. If you feel tempted by them as 'easier' wins, note they are the identical half-of-fair pricing - a shorter route to the same 50% edge.

A worked example

You pick 1-1-2-3. Repeats: the 1 appears twice, so ways = 4!/2! = 12 - a 12-way ticket.

Box payout = $5,000 / 12 = $416.67. Box probability = 12/10,000 = 3/2,500. Straight probability = 1/10,000.

EV audit: Straight (1/10,000) x $5,000 = $0.50. Box (12/10,000) x $416.67 = $0.50. Pair (your 1-1 in order) (1/100) x $50 = $0.50. Front 3 (1-1-2) (1/1,000) x $500 = $0.50. House edge 50% on each.

The draw comes 1-3-1-2. Straight misses. Box hits - the drawn multiset {1,1,2,3} matches yours - paying $416.67. Pair misses (drawn 1-3 vs your 1-1). Front 3 misses (1-3-1 vs 1-1-2). Before the draw, all of that upside was worth exactly $0.50 per play type per dollar.

Common mistakes

• Treating the $5,000 jackpot as evidence Pick 4 beats Pick 3. Ten times the payout at a tenth the probability is the same $0.50 EV.

• Miscounting ways on double pairs: 1-1-2-2 is 4!/(2!2!) = 6, not 12.

• Reading Front 3 as needing any 3 matching digits. It is the FIRST three digits in exact order - a strict positional match.

• Assuming a Box hit excludes other wins. One draw is checked against every play type; a Straight also pays Pair and both Front/Back 3 when the digits line up.

• Hunting for the play type with the best odds. The last question in the set exists to make you prove there is not one: every option prices at 1 - EV = 50%.

Why interviews test this

Pick 4 is where interviewers escalate the counting: permutations of multisets (the 12-way, 6-way, 4-way cases) are a staple combinatorics screen, and the follow-up 'so which bet is best?' tests whether the candidate has internalized that probability times payout, not either factor alone, is the object of interest.

The invariant - every bet priced at exactly half of fair - is also a miniature of how real derivative pricing questions work: given odds, compute the fair value, compare to the offered price, and state the edge. Here the edge is minus 50% everywhere, and the only winning move is the lab's title.