Pick 4 Odds Calculator

Pick 4 is four digits, 0000–9999, 10,000 equally likely ordered outcomes. Defaults: Straight, $1. Calculate. Odds = 1:10,000. Typical Payout = $5,000. Fair Payout = $1/0.0001 − $1 = $9,999. Expected value at the typical chart is 0.0001×5000 − 1 = −$0.50, the same 50 percent stake take as default Pick 3 straight.

Play Type enumerates the box classes by permutation count of the digit multiset. 4-Way Box: three-of-a-kind plus a singleton (1112 and its four positions for the 2), p = 4/10,000, typical $1,200, fair $2,499. 6-Way Box: two pair (1122), 4!/(2!2!)=6 permutations, typical $800, fair ≈ $1,665.67. 12-Way Box: one pair and two distinct other digits (1123), 4!/2! = 12, typical $400, fair ≈ $832.33. 24-Way Box: four distinct digits (1234), 4! = 24, typical $200, fair ≈ $415.67. Straight remains the exact order, 1/10,000, typical $5,000.

Pick the box that matches your multiset. 1234 is 24-way, not 12-way. 1123 is 12-way. 1122 is 6-way. 1112 is 4-way. 1111 is straight-only (one permutation). The calculator will apply whichever type you select even if it disagrees with the digits you have in mind — there is no digit-entry field, only the type. That is deliberate: you are pricing a class of bets, not validating a slip.

Compare Fair and Typical: 24-way typical $200 versus fair $415.67 is about a 52 percent haircut on fair profit; 4-way $1,200 versus $2,499 is similar. Charts differ by state; some pay $9,000 straight on $1, which is a different product. Use your commission’s published payoff as a mental override of “typical.” Wager scales dollars linearly; odds stay put.

Wheel-style “all permutations as straights” on 1234 is 24 straight tickets, not one 24-way box: you would pay $24 and collect a straight payoff if any order hits, which is a different EV than one boxed $1. Run both stories if you are deciding how to write the slip. Pair with Pick 3 if you play both daily numbers; the sample spaces (1,000 vs 10,000) do not combine into a single hypergeometric.

Pick 4 extended the daily numbers format from three digits to four when states wanted a larger prize without moving to a 6/49 matrix. The combinatorial flavour is still digits with replacement, not lotto combinations. Illegal four-digit numbers games existed; legal Pick 4 industrialised them with official randomisation and posted pay tables. Straight, boxed, pairs, and combination plays are the menu. SorteCalc implements the permutation-count boxes plus straight, which covers the bets whose odds are ways/10,000.

The five typical payoffs — 5000, 1200, 800, 400, 200 — are a teaching ladder, not a regulator filing. They sit near half the fair gross, so the house keeps about half the stake across box types. That uniformity is intentional design: players should not be able to shop a much better RTP by switching from straight to 24-way. If your state posts a 24-way payoff that implies a materially different EV than straight, that is a chart quirk worth computing, not a hidden edge that lasts.

Digit multisets are classified by partition of the integer 4: 4 (quads), 3+1 (trips), 2+2 (two pair), 2+1+1 (one pair), 1+1+1+1 (all distinct). Those partitions are exactly 1-way, 4-way, 6-way, 12-way, 24-way. Learning to see 1123 as the partition 2+1+1 is the only skill the box menu requires. Everything else is 10,000 equally likely strings.

Relative to Pick 3, wins are rarer and payoffs larger; the default edge is similar. Relative to 6/49, you still have a daily cadence and a posted prize, not a rolling jackpot. There is no “due” number: 1234 is as likely tomorrow as 0000. Hot-and-cold digit sheets are a sideshow. The law of truly large numbers says someone hits 4-digit straights every day in a large state; it does not say your 24-way box is catching up after a losing month.

Use this calculator to read a pay table. Convert each published prize into EV = (ways/10000)×prize − stake and compare across types. If they cluster around −$0.50 on a $1 bet, you are looking at a standard numbers product. If a promotional multiplier weekend changes one rung, recompute that rung only. Then decide whether a 50 percent expected loss per dollar is an acceptable price for a daily ritual.

Pick 4 extends the daily numbers format from three digits to four when a state wants a larger posted prize without moving to a six-from-forty-nine matrix. The sample space is still digits with replacement, not lotto combinations: four ordered positions, each 0 through 9, ten thousand equally likely strings 0000 through 9999. Straight play is one of those strings. Probability 1 / 10,000. Typical teaching payout 5,000 dollars on a one-dollar stake. Fair net profit is 9,999 dollars, the 9999-to-1 line. Expected value at the typical chart is 0.0001 times 5000 minus 1, again minus 0.50 dollars, the same 50 percent stake take as default Pick 3 straight.

Box classes follow the integer partitions of 4, which are the permutation counts of the digit multiset. Four-way box is three-of-a-kind plus a singleton (1112 and its four positions for the 2), probability 4 / 10,000, typical 1,200, fair 2,499. Six-way box is two pair (1122), 4 factorial over 2 factorial times 2 factorial equals 6 permutations, typical 800, fair about 1,665.67. Twelve-way box is one pair and two distinct other digits (1123), 4 factorial over 2 factorial equals 12, typical 400, fair about 832.33. Twenty-four-way box is four distinct digits (1234), 4 factorial equals 24, typical 200, fair about 415.67. Straight remains exact order.

Expected values on that teaching ladder sit together. Straight: 5000 / 10,000 minus 1 equals minus 0.50. Four-way: 4 times 1200 / 10,000 minus 1 equals minus 0.52. Six-way, twelve-way, and twenty-four-way at 800, 400, and 200 similarly yield minus 0.52. Players should not be able to shop a much better return by switching from straight to twenty-four-way. If a state posts a twenty-four-way payoff that implies a materially different mean, that is a chart quirk worth computing, not a hidden edge that lasts.

Permutation origins are the whole skill the box menu requires. Four distinct digits, 24 orders. Pattern aabc, 12 orders. Pattern aabb, 6. Pattern aaab, 4. Pattern aaaa, 1. Those way-counts are the size of the winning set for one boxed ticket, not the number of such boxes in the universe. Census: C(10, 4) times 24 equals 5,040 all-distinct ordered tuples, and the remaining partitions fill the 10,000. Pick 1234 as twenty-four-way, not twelve-way. Pick 1123 as twelve-way. Pick 1111 as straight-only.

Buying all 24 straights on 1234 is not one twenty-four-way box. You would pay 24 dollars and collect a straight prize if any order hits, which is a different mean: probability 24 / 10,000 times 5,000 minus 24 equals minus 12 dollars at the typical straight chart, worse dollar expected value than one boxed dollar at minus 0.52 because you over-bought. The box exists to sell the permutation set as one ticket at a lower prize.

Assumptions: uniform 10,000, teaching typicals 5000 / 1200 / 800 / 400 / 200, fair column as net at zero mean. There is no hypergeometric pool-and-draw and no bonus drum. Hot-and-cold digit sheets are a sideshow; 1234 is as likely tomorrow as 0000. A year of daily straights has at-least-once chance 1 minus (1 minus 1/10,000) to the 365, about 3.58 percent, and expected loss 365 times 0.50 dollars at the default chart. Read your commission’s published payoff as an override of typical. Then decide whether a 50 percent expected loss per dollar is an acceptable price for a daily ritual.