7. A fair coin is tossed in sequence 5 times and it is recorded whether heads or tails has landed (e.g., HTHTH). What is the probability that heads will land at least three times in a row?
7. The total number of possible outcomes is $2 \cdot 2 \cdot 2 \cdot 2 \cdot 2=32$.

1 POINT

Let $A$ be the event that at least three heads appear in a row.

Thus, we are interested in cases where either 3, 4, or 5 heads appear in a row. 1 POINT

5 heads in a row ( HHHHH ) - 1 favorable outcome. 1 POINT

4 heads in a row ( THHHH, HHHHT ) - 2 favorable outcomes. 2 POINTS

3 heads in a row ( HHHHT, THHHT, TTHHH, HTHHH, HHHHT ) - 5 favorable outcomes.

3 POINTS

The desired probability is $P(A)=\frac{1+2+5}{32}=\frac{8}{32}=\frac{1}{4}$.

2 POINTS