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In a “6-from-49” lottery, a player selects \(6\) distinct numbers, and the lottery draws \(6\) distinct winning numbers from \(1\) through \(49\). Order does not matter.
1. Find the theoretical probability that exactly \(3\) of the player's numbers match the winning numbers.
2. In a computer simulation of \(1{,}000{,}000\) drawings, exactly \(3\) numbers matched in \(17{,}612\) trials. Find the relative frequency and its absolute difference from the theoretical probability.
3. Explain how the simulation result relates to the theoretical probability under the law of large numbers.
Hints
- Count unordered selections with combinations.
- For exactly \(3\) matches, choose matched numbers from the player's selections and unmatched numbers from the remaining numbers.
- Relative frequency is the number of successful trials divided by the total number of trials.
Solution
1. There are \(\binom{49}{6}=13{,}983{,}816\) possible drawings. To obtain exactly \(3\) matches, choose \(3\) of the player's \(6\) numbers and \(3\) of the other \(43\) numbers. Thus, \(P(\text{exactly 3 matches})=\frac{\binom{6}{3}\binom{43}{3}}{\binom{49}{6}}=\frac{246820}{13983816}\approx 0.0176504\).
2. The relative frequency is \(\frac{17612}{1000000}=0.017612\). The absolute difference is \(\left|0.017612-0.0176504\right|\approx 0.0000384\), or about \(0.00384\) percentage points.
3. The law of large numbers states that, as the number of independent trials increases, the relative frequency tends to stabilize near the theoretical probability. The small difference after one million trials is consistent with this behavior.
Answer
1. \(P(\text{exactly 3 matches})\approx 0.0176504\), or about \(1.765\%\)
2. Relative frequency \(0.017612\); absolute difference approximately \(0.0000384\), or \(0.00384\) percentage points
3. With many trials, the relative frequency tends to approach the theoretical probability.
