Problem 9 · AMC 8 Stretch
Stretch
Counting & Probability
accounting-for-all-possibilitieslogical-reasoning
A small lottery puts 10 balls numbered 1 to 10 in a bag, then draws 3 of them one at a time. To win, the 3 numbers drawn must match the 3 numbers you bet on (order doesn't matter). What is the probability you win? (Then think: is a big real lottery, like 'pick 6 numbers out of 50,' a good way to spend your money?)
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Answer: \(\frac{1}{120}\) (the small lottery); a real 'pick 6 of 50' lottery is about 1 in 16 million
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Hint 1 of 4
Draw the balls one at a time. What is the chance the first ball drawn is one of your 3 numbers, out of the 10 in the bag?
Still stuck? Show hint 2 →
Hint 2 of 4
1st ball: \(\frac{3}{10}\) chance it's one of yours. Then 9 balls remain and 2 of your numbers are left, so the 2nd ball: \(\frac{2}{9}\). Keep going.
Still stuck? Show hint 3 →
Hint 3 of 4
Multiply the three fractions: \(\frac{3}{10}\times\frac{2}{9}\times\frac{1}{8}\).
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Approach: Multiplication principle — multiply the shrinking match-fractions
- Draw the 3 balls one at a time, and each one must be one of your numbers that hasn't been matched yet.
- 1st ball is one of your 3 numbers, out of 10 balls: \(\frac{3}{10}\). 2nd ball is one of your remaining 2 numbers, out of the 9 left: \(\frac{2}{9}\). 3rd ball is your last number, out of the 8 left: \(\frac{1}{8}\).
- Multiply: \(\frac{3}{10}\times\frac{2}{9}\times\frac{1}{8}=\frac{6}{720}=\frac{1}{120}\). So even in this tiny lottery your chance is only 1 in 120.
- A real 'pick 6 of 50' lottery uses the same idea: \(\frac{6}{50}\times\frac{5}{49}\times\frac{4}{48}\times\frac{3}{47}\times\frac{2}{46}\times\frac{1}{45}\approx\frac{1}{15{,}900{,}000}\), about 1 in 16 million. The lesson: the lottery is a very poor investment.
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