Problem
Suppose Dan wins $40 \%$ of all staring contests. (a) What is the probability that Dan wins two staring contests in a row? (b) What is the probability that Dan wins six staring contests in a row? (c) When events are independent, their complements are independent as well. Use this result to determine the probability that Dan wins six staring contests in a row, but does not win seven in a row. (a) The probability that Dan wins two staring contests in a row is (Round to four decimal places as needed.)
Solution
Step 1 ：Suppose Dan wins 40% of all staring contests.
Step 2 ：The probability of independent events occurring in sequence is the product of their individual probabilities. Since the probability of Dan winning a single staring contest is 40%, the probability of him winning two in a row would be \(0.4 * 0.4\).
Step 3 ：So, the probability that Dan wins two staring contests in a row is \(0.4 * 0.4 = 0.16\).
Step 4 ：Final Answer: The probability that Dan wins two staring contests in a row is \(\boxed{0.16}\).
