Problem
Once a woman won $\$ 1$ million in a scratch-off game from a lottery. Some years later, she won $\$ 1$ million in another scratch-off game. In the first game, she beat odds of 1 in 4.2 million to win. In the second, she beat odds of 1 in 805,600 . (a) What is the probability that an individual would win $\$ 1$ million in both games if they bought one scratch-off ticket from each game? (b) What is the probability that an individual would win $\$ 1$ million twice in the second scratch-off game? (Use scientific notation. Use the mêltiplication symbol in the math palette as needed. Round to the nearest tenth as needed.) (b) The probability that an individual would win $\$ 1$ million twice in the second scratch-off game is (Use scientific notation. Use the multiplication symbol in the math palette as needed. Round to the nearest tenth as needed.)
Solution
Step 1 ：The probability of winning in each game is the reciprocal of the odds given. So, the probability of winning the first game is \(\frac{1}{4.2 \times 10^{6}}\) and the probability of winning the second game is \(\frac{1}{805,600}\).
Step 2 ：The probability of two independent events both happening is the product of their individual probabilities.
Step 3 ：So, the probability of winning both games is \(\frac{1}{4.2 \times 10^{6}} \times \frac{1}{805,600}\).
Step 4 ：Calculating the above expression, we get approximately \(2.96 \times 10^{-13}\).
Step 5 ：Final Answer: The probability that an individual would win $1 million in both games if they bought one scratch-off ticket from each game is approximately \(\boxed{2.96 \times 10^{-13}}\).
