Other: Blocks on the Streets

A student leaves University (\(U\)) to walk Home (\(H\)), a distance of four blocks in a straight line. At each crossing, they toss a fair coin to decide whether to move forward (Heads) or backward (Tails). Note that from \(U\) they cannot go backwards, so a Tail means they stay put. If it takes 10 minutes to walk each block, what is the probability they will be home in less than 45 minutes?

Answer

The key observation is that the first move from \(U\) is always forward, effectively requiring only 3 additional Heads to reach home within 4 tosses (since \(4 \times 10 = 40\) minutes \(< 45\) minutes). Therefore, the probability of reaching home in less than 45 minutes is the probability of getting 3 consecutive Heads:

\[ \mathbb{P}(\text{3 Heads}) = \left(\tfrac{1}{2}\right)^3 = \tfrac{1}{8}. \]
Back to collection