To determine the probability of winning a super prize in the SuperLottery, we begin by calculating the total number of possible outcomes for drawing the balls and then the outcomes that lead to winning conditions.
Total Outcomes
-
Drawing the White Balls: The three white balls are drawn without replacement from a total of 10 white balls (numbered from 1 to 10). The number of ways to choose 3 balls from 10 is given by the combination formula: \[ \binom{10}{3} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120. \]
-
Drawing the SuperBall: Next, the SuperBall is drawn from a set of 10 red balls (numbered from 11 to 20). There are 10 possible choices for the SuperBall.
Thus, the total number of possible lottery outcomes (combinations of white balls and one SuperBall) is: \[ \text{Total Outcomes} = \binom{10}{3} \times 10 = 120 \times 10 = 1200. \]
Winning Conditions
The winning conditions stipulate that a ticket wins if it matches at least two of the white balls or the red SuperBall. We will first calculate the outcomes for matching at least 2 white balls.
Winning by Matching 2 or 3 White Balls or Matching the SuperBall
-
Case A: Matching 3 White Balls:
- Choose 3 white balls that match (1 way to choose these).
- The SuperBall can be anything (10 options). Thus, for matching all 3: \[ 1 \times 10 = 10 \text{ outcomes}. \]
-
Case B: Matching Exactly 2 White Balls:
- Choose 2 white balls from your ticket (which have been chosen by the player) matching the drawn balls: \[ \binom{3}{2} = 3 \text{ (the player can choose 2 out of their 3 numbers)}. \]
- Choose 1 white ball from the remaining 7 (to make a total of 3 drawn): \[ \binom{7}{1} = 7. \]
- The SuperBall can be anything: \[ 10 \text{ options for SuperBall}. \] Thus, for matching exactly 2: \[ 3 \times 7 \times 10 = 210 \text{ outcomes}. \]
-
Case C: Matching the SuperBall:
- The SuperBall matches (1 way to choose), with no restrictions on white balls.
- From 10 white balls, the player could have:
- 0 matches: \(\binom{7}{3} = 35\) ways,
- 1 match: \(\binom{3}{1} \times \binom{7}{2} = 3 \times 21 = 63\) ways,
- 2 matches: \(\binom{3}{2} \times \binom{7}{1} = 3 \times 7 = 21\) ways,
- 3 matches: 1 way. Calculating total from this case: \[ 1 \times (35 + 63 + 21 + 1) = 1 \times 120 = 120 \text{ outcomes}. \]
Total Winning Outcomes
-
The total winning outcomes that come from matching at least 2 white balls (10 for 3 matches and 210 for 2 matches) can overlap with the outcomes from matching the SuperBall. Thus, we subtract the outcomes counted in both cases (matching exactly 2 white balls alongside the SuperBall):
Outcomes counted in both:
- Matching exactly 2 White Balls (210) and also matching the SuperBall (1).
Thus, we can apply the inclusion-exclusion principle: \[ \text{Winning Outcomes} = (\text{matches 3 white}) + (\text{matches exactly 2 white}) + (\text{matches SuperBall}) - (\text{overlap matches SuperBall and 2 white}). \]
Therefore: \[ \text{Winning Outcomes} = 10 + 210 + 120 - 21 = 319. \]
Probability of Winning a Super Prize
Finally, we can now calculate the probability: \[ \text{Probability of Winning} = \frac{\text{Winning Outcomes}}{\text{Total Outcomes}} = \frac{319}{1200}. \]
Thus, the probability of winning a super prize is: \[ \boxed{\frac{319}{1200}}. \]