Probability of Lotto (3/50 + 2 Bonus)

Analysis of winning chances in "3 out of 50" lottery with 2 bonus numbers over 20 years

Cumulative Probability
Probability per Draw

Mathematical Calculation

Probability of Lotto
Getting 3 correct numbers with a bonus number in one draw:

P = 5 3 × 45 2 × 2 1 × 10 1 ÷ [ 50 5 × 12 2 ]

Where:

P = (10 × 990 × 2 × 10) / (2,118,760 × 66) = 198,000 / 139,838,160 = 0.001416 (0.1416%)

Probability Over Time

Cumulative probability over 20 years with two plays per week:

  • After 5 years (520 draws): 51.5%
  • After 10 years (1040 draws): 76.5%
  • After 15 years (1560 draws): 89.0%
  • After 20 years (2080 draws): 94.7%

Probability increases steadily and reaches 94.7% after 20 years.

Interpretation

The chart shows two lines:

  • Blue line: Cumulative probability of at least one win
  • Red line: Constant probability per draw (0.1416%)

With regular play over 20 years, the chance is:

P(at least 1 win) = 1 - (1 - 0.001416)2080 = 0.947 (94.7%)

This means about 95 out of 100 players would achieve this win at least once during this period.

Economic Considerations

At a cost of $2.50 per play and two plays per week:

The average payout for "3 correct + bonus number" is about $50-100. Even with multiple wins over 20 years, the overall balance is likely negative.