Double Jackpot Shawn lives near the border of Illinois and Missouri. One weekend he decides to play \$1 in both state lotteries in hopes of hitting two jackpots. The probability of winning the Missouri Lotto is about 0.00000028357 and the probability of winning the Illinois Lotto is about 0.000000098239.
b. Find the probability that Shawn will win both jackpots.

Verified step by step guidance

1. Identify the probabilities given: the probability of winning the Missouri Lotto is \(p_M = 0.00000028357\) and the probability of winning the Illinois Lotto is \(p_I = 0.000000098239\).
2. Since the two lotteries are independent events (winning one does not affect winning the other), the probability of both events happening together is the product of their individual probabilities.
3. Write the formula for the probability of both independent events occurring: \(P(\text{both}) = p_M \times p_I\).
4. Substitute the given values into the formula: \(P(\text{both}) = 0.00000028357 \times 0.000000098239\).
5. Multiply the two probabilities to find the probability that Shawn will win both jackpots.

Key Concepts

Independent Events

Two events are independent if the occurrence of one does not affect the probability of the other. In this problem, winning the Missouri Lotto and winning the Illinois Lotto are independent events because the outcome of one lottery does not influence the other.

Multiplication Rule for Independent Events

The probability of two independent events both occurring is the product of their individual probabilities. To find the chance of winning both jackpots, multiply the probability of winning the Missouri Lotto by the probability of winning the Illinois Lotto.

Probability Values and Interpretation

Probabilities range from 0 to 1, where values close to zero indicate very unlikely events. Understanding the extremely small probabilities given helps interpret how rare it is to win either lottery, and especially both simultaneously.