Compute exact odds, expected value, and prize-tier probabilities for any lottery format. Supports 6/49, Powerball, Mega Millions, and custom games. Visualize your chances with interactive charts and learn the mathematics behind lottery games.
The Lottery Probability Calculator is designed to demystify the odds behind popular lottery games. Whether you are playing Powerball, Mega Millions, or a local 6/49 draw, the underlying mathematics is rooted in combinatorics and probability theory. This tool computes exact probabilities for each prize tier, the overall chance of winning, and the expected value — a key metric that reveals the true cost of playing.
The number of ways to choose k numbers from a pool of N is:
C(N, k) = N! / (k! · (N−k)!)
For a game with a bonus pool of M numbers and b drawn, the total combinations are C(N, k) × C(M, b).
To win the jackpot in a typical 6/49 game, you must match all 6 numbers drawn from a pool of 49. The total number of possible combinations is C(49, 6) = 13,983,816. Therefore, the probability of winning the jackpot is 1 in 13,983,816 — approximately 0.00000715%. For Powerball, the math is even more staggering: you must match 5 numbers from 69 and the Powerball from 26, giving C(69, 5) × 26 = 292,201,338 possible combinations. The chance of winning the Powerball jackpot is about 1 in 292 million.
The calculator also computes lower-tier probabilities. For example, matching 5 numbers (but not the Powerball) has its own probability, calculated by counting the number of ways to choose 5 correct numbers from the 5 drawn and 0 from the remaining 64, multiplied by the ways to miss the Powerball. These calculations are performed using hypergeometric distributions, which model drawing without replacement.
The overall probability of winning any prize is the sum of the probabilities of all prize tiers. In many lotteries, this ranges from 1 in 24 to 1 in 40, depending on the prize structure. However, the overwhelming majority of wins are small prizes (e.g., a free ticket or a few dollars), which is why the expected value remains negative for players.
Expected value (EV) is the average amount you can expect to win (or lose) per ticket over the long run. It is calculated as:
EV = Σ (Prize_i × Probability_i) − Ticket Price
For example, if a jackpot of $10 million has a probability of 1 in 292 million, its contribution to EV is about $0.034. Adding all prize tiers typically yields an EV of −$1.00 to −$1.50 for a $2 ticket, meaning that for every $2 you spend, you expect to lose about $1.00–$1.50 in the long run. This is why lotteries are often described as a "voluntary tax" — the house edge is built into the odds.
It is important to note that the jackpot size affects EV. When the jackpot grows to record levels (e.g., $1 billion), the EV can become positive for a single drawing. However, this does not account for taxes, the possibility of shared jackpots, or the time value of money for annuity payments. The calculator allows you to input your own prize amounts to see how changes affect the expected value.
Lotteries have a long history, dating back to ancient China and Rome, where they were used to fund public projects. The modern lottery as we know it emerged in the 20th century, with the first modern state lottery in the United States established in New Hampshire in 1964. Today, lotteries generate billions of dollars in revenue worldwide, with a significant portion allocated to education, infrastructure, and social programs.
Despite the overwhelming odds, people continue to play because of the psychology of gambling. The "near-miss" effect, the excitement of the draw, and the fantasy of sudden wealth all contribute to the persistent appeal. Behavioral economists have shown that people systematically overestimate small probabilities, a phenomenon known as probability weighting. This calculator aims to provide objective, data-driven insights to help you make informed decisions.
The lottery should be viewed as entertainment, not as an investment or a reliable path to wealth. The odds are overwhelmingly against you, and the expected value is negative. Set a budget for how much you are willing to spend, and never chase losses. If you or someone you know is struggling with gambling addiction, seek help from professional organizations such as the National Council on Problem Gambling (1-800-522-4700) or Gamblers Anonymous.
This calculator is designed to educate and empower you with accurate information. Understanding the true odds and expected value can help you make rational decisions about whether and how much to play.
In November 2022, the Powerball jackpot reached a record $2.04 billion. At that level, the expected value of a $2 ticket (ignoring taxes and shared jackpots) was approximately $2.32 — positive for the first time in history. However, after accounting for federal and state taxes (which can take up to 40% of the winnings) and the probability of splitting the jackpot, the EV quickly dropped back into negative territory. This case illustrates how the calculator can be used to analyze real scenarios and understand the impact of taxes, sharing, and annuity choices on the true value of a ticket.