Goal‐Scoring Streak
Imagine a young soccer enthusiast, Alex, who has an 80% chance of scoring a goal on each kick, with each attempt being independent. What is the expected number of consecutive attempts Alex needs to make in order to score two goals in a row?
Answer
Using the Law of Total Expectation, we have:
\[
E_0 = 0.8\,(1+E_1) + 0.2\,(1+E_0),
\quad
E_1 = 0.8\cdot1 + 0.2\,(1+E_0).
\]
Solving gives
\[
E_0 = \frac{45}{16}.
\]