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? (a) The probability that an individual would win $\$ 1$ million in both games if they bought one scratch-off ticket from each 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 the first game is 1 in 4.2 million, which can be written as \(2.380952380952381 \times 10^{-7}\).
Step 2 ：The probability of winning the second game is 1 in 805,600, which can be written as \(1.2413108242303872 \times 10^{-6}\).
Step 3 ：The probability of both events happening is the product of their individual probabilities. Therefore, we multiply the two probabilities together to get \(2.9555019624533027 \times 10^{-13}\).
Step 4 ：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 \(\boxed{2.96 \times 10^{-13}}\).
