Pick 3 Odds Calculator

Pick 3 is a three-digit daily numbers game: 000 through 999, one thousand equally likely outcomes if each digit is 0–9 with replacement (or three independent 0–9 balls, same counting). Defaults: play type Straight, wager $1. Straight means you named the exact order. Calculate. Odds = 1:1,000. Typical Payout = $500 on a $1 stake (many US lotteries’ classic straight payoff). Fair Payout = wager/p − wager = $1/0.001 − $1 = $999, the profit that would make the bet even-money in expectation.

Switch Play Type to 3-Way Box. That is for a number with a pair, like 112, 121, or 211 — three permutations. Odds = 3/1000 = 1:333. Typical Payout ≈ $160 on $1. Fair Payout = 1/0.003 − 1 ≈ $332.33. Switch to 6-Way Box for three distinct digits (123 and its six orders). Odds = 6/1000 = 1:167. Typical Payout ≈ $80. Fair Payout ≈ $165.67. The typical column is a representative published payoff, not your state’s exact chart; the fair column is mathematics.

The 3-way option is invalid for a 6-way number and vice versa. If you box 123 as 3-way, you have mis-set the type; the calculator will still apply 3/1000, which is the wrong model. Straight on 112 still pays only if the draw is exactly the order you wrote. Boxing is how you buy the permutation set.

Raise Wager to $2 to double both typical and fair dollars; odds do not change. Some states sell $0.50 straights at half the $1 payoff. House edge on the default straight is 1 − 500/999 ≈ 50 percent of the fair profit, or more cleanly: expected return = 0.001 × $500 − $1 = −$0.50, a 50 percent take on the stake. That is a harsher edge than a typical 6/49 ticket’s combined prize fund, and it is why Pick 3 is a daily cash cow.

Use Straight when you have an exact order (or when you buy all orders as separate straights). Use 3-Way Box only for paired digits. Use 6-Way Box for three unique digits. Then look at Fair versus Typical: the gap is the product. This is not a parimutuel jackpot; it is a fixed-odd numbers game descended from the illegal daily numbers racket.

Before state lotteries, the numbers game — policy, bolita, the daily number — was an illegal urban fixture in the United States, with 3-digit outcomes taken from published figures (pari-mutuel handle digits, Treasury balances, later the New York Clearing House). Payoffs were roughly 600-to-1 on a 999-to-1 fair bet, with runners and bankers taking the rest. When states legalised Pick 3 / Daily Numbers in the 1970s and 1980s, they kept the format and often the 500-for-1 straight tradition, converting a black market into a public-finance instrument.

Straight, box, wheel, pairs, and front-or-back plays are all ways of buying subsets of the 1,000 outcomes. A 6-way box on 123 is the set {123,132,213,231,312,321}. A 3-way box on 112 is {112,121,211}. A straight is a singleton. Combinatorics of digit multisets is the whole subject; there is no hypergeometric urn of 49 balls here. Each digit is typically an independent 0–9 draw, so 10^3 = 1,000, not C(10,3).

Typical payoffs (500 / 160 / 80 on a $1 bet) are in the neighbourhood of a 50 percent return-to-player. Exact charts vary by lottery: some pay $500 straight, some $600, some include fireballs and multipliers that change the distribution without fixing the edge. SorteCalc’s “typical” column is a teaching default aligned with common US charts, not a promise that your commission pays $500. The fair column is the unique payoff that sets EV = 0 at the stated odds.

Compared with 6/49, Pick 3 has frequent small wins, daily draws, and a brutal house edge. That combination is catnip for habit and terrible for expected value. Clotfelter and Cook noted that numbers games skim heavily because the probability is legible (1 in 1,000 sounds almost sports-book-like) while the payoff is not fair. A sports bet at 1 in 1000 would not pay 500-to-1 if a book were posting an honest line; it would pay near 999-to-1 minus a few percent juice.

SorteCalc does not simulate fireball, exact-order plus box combos, or midday/evening separate draws beyond treating each draw as another independent 1/1000. Stack draws with the multi-ticket logic if you play every day: 1 − (1−0.001)^365 ≈ 30.6 percent chance of at least one straight hit in a year of daily singles — and expected loss 365 × $0.50 = $182.50 at a $500 payoff. Play it as a priced habit or do not play it.

Before state lotteries, the numbers game — policy, bolita, the daily number — was an illegal urban fixture in the United States, with three-digit outcomes taken from published figures. Payoffs were roughly 600-to-1 on a 999-to-1 fair bet. When states legalised Pick 3 they kept the format and often the 500-for-1 straight tradition. The sample space is not a lotto urn. It is the product set of three digits, each 0 through 9, one thousand equally likely ordered triples 000 through 999.

Straight play names one ordered triple. Probability is 1 / 1000. Fair profit on a one-dollar stake is stake over p minus stake, which is 999 dollars: that is the 999-to-1 net that would make expected value zero. Typical posted payout is 500 dollars on a one-dollar straight, a house chart not a fair chart. Expected value at that typical is 0.001 times 500 minus 1, which is minus 0.50 dollars, a 50 percent take on the stake. That is harsher than a typical six-from-forty-nine prize fund and it is why a daily numbers game is a cash cow.

Three-way box is the paired-digit multiset, pattern aab, three permutations, probability 3 / 1000. Fair profit is (1000 / 3 minus 1) dollars, about 332.33. Typical posted payout is about 160 dollars on a dollar. Six-way box is three distinct digits, 3 factorial equals 6 permutations, probability 6 / 1000. Fair profit about 165.67. Typical posted payout about 80 dollars. The 160 and 80 figures are set so that boxed expected value sits near minus 52 percent of stake, in the same neighbourhood as the straight house take, not at the fair 999-to-1 line.

Permutation census audits the thousand. Ten triples of the form aaa. Two-hundred seventy ordered triples with a pair (10 choices for the paired digit times 9 for the singleton times 3 positions for the singleton). Seven-hundred twenty ordered triples with three distinct digits (10 times 9 times 8). The sum is 1,000. Number of six-way boxes is C(10, 3) = 120, and 120 times 6 equals 720. Number of three-way boxes is 10 times 9 equals 90, and 90 times 3 equals 270. A triple such as 777 is straight-only; do not apply three-way arithmetic to it. Boxing 123 as three-way is the wrong model.

Order versus combination is the operator’s rule, not a folk identity. Straight is ordered; box is the unordered multiset. Players sometimes think 123 is “one combination” and should pay six-way even on a straight ticket. The posted chart decides. Digits are independent: 1-2-3 is as likely as 7-7-7. There is no hypergeometric n-and-k here and no bonus drum. Wager scales dollars linearly; odds stay 1 in 1000 over the way-count.

Assumptions: uniform 000 to 999, fixed typical prizes 500 / 160 / 80 as a teaching chart not your commission’s filing, fair column defined as net profit at zero expected value. Fireball add-ons, front-or-back pairs, and midday versus evening as separate drawings are extra independent trials at the same p. Stack a year of daily singles with the complement formula if you like: 1 minus (0.999) to the 365 is about 30.6 percent chance of at least one straight hit, and expected loss 365 times 0.50 dollars at the 500 chart. Priced habit or not a habit.