How Big Is Half a Lottery?

Lottery fever has swept through your town. The regional lottery is advertising a $1 billion jackpot — one prize, one winner.

Each ticket is a sequence of seven numbers, drawn one at a time with replacement from a bin containing balls labeled 1 through 19. Because each ball is returned to the bin before the next draw, the same number may appear more than once on a single ticket. All 19⁷ possible sequences are equally likely to be chosen as the winning ticket.

You may purchase as many distinct tickets as you wish at $1 each. You win the jackpot if any one of your tickets exactly matches the winning sequence.

The challenge

How many tickets must you purchase so that your probability of holding the winning ticket is at least 50%?

Express your answer as a whole number and show the inequality you solved to find it.

Interactive Supplement

Lottery Expected Value Explorer — Interactive

Explore this puzzle visually with an interactive diagram — drag sliders, watch the geometry update in real time, and build intuition before you solve.

Open interactive →

💡 Hint

Start by counting the total number of distinct tickets that could ever be printed. Because draws are independent with replacement, each of the seven positions can hold any of the 19 values independently of the others.

Once you have that count, think about what it means to hold k distinct tickets. Your tickets cover exactly k of the possible outcomes. Since the winning ticket is drawn uniformly at random from all possible outcomes, the probability that one of your k tickets is the winner is simply a ratio.

Set that ratio greater than or equal to 1 / 2 and solve for k. Because k must be a whole number, you will need to take a ceiling.

Solution

Step 1 — Count the total number of possible tickets

Each ticket is a sequence of 7 numbers, each chosen independently from {1, 2, …, 19}. Because the draws are with replacement, each position has 19 independent choices. By the multiplication principle:

N = 19⁷ = 893,871,739

There are exactly 893,871,739 distinct tickets.

Step 2 — Set up the probability inequality

Suppose you purchase k distinct tickets. Each covers exactly one outcome. The winning ticket is drawn uniformly at random from all N outcomes, so:

P(win) = k / N

You want this probability to be at least 1 / 2:

k / 893,871,739 ≥ 1/2

Step 3 — Solve for k

Multiplying both sides by N:

k ≥ 893,871,739 / 2 = 446,935,869.5

Since k must be a whole number (you cannot buy half a ticket), take the ceiling:

k = 446,935,870 tickets

Minimum tickets for a 50% chance of winning

The uncomfortable punchline

Buying 446,935,870 tickets costs exactly $446,935,870. Even if you win, the $1 billion advertised jackpot is paid as a lump sum of roughly $600 million (about 60% of the advertised amount), and after federal income tax at 37% that leaves approximately $378 million in hand.

Your expected net profit at the 50% threshold is therefore:

E[profit] = 0.5 × $378,000,000 − $446,935,870 ≈ −$257,935,870

You expect to lose over $257 million even when buying exactly enough tickets for a coin-flip chance at the jackpot. No number of tickets makes this lottery profitable — the net jackpot ($378M) is less than half the total ticket space ($893.9M), so the expected value is negative for every possible purchasing strategy. Use the interactive diagram to verify this for any number of tickets you choose.

This result generalises: a lottery is only ever positive expected value when the after-tax lump-sum jackpot exceeds the total size of the ticket space. That threshold here would require a net payout of at least $893,871,739 — more than twice the actual after-tax amount.