Question 99
Alberto and Barbara play table football. Before starting, they decide that the game will end as soon as one of them has scored 3 goals more than the other. Knowing that for each ball played both Alberto and Barbara have a $50\%$ probability of scoring, what is the probability that the game does not end before the twenty-first goal?
(A) $\frac{3^{9}}{2^{18}}$
(B) $\frac{3^{10}}{2^{20}}$
(C) $\frac{3^{20}}{2^{40}}$
(D) $\frac{13 \cdot 17 \cdot 19 \cdot 31}{2^{18}}$
(E) $\frac{3 \cdot 7 \cdot 13 \cdot 17 \cdot 19}{2^{18}}$
(A) $\frac{3^{9}}{2^{18}}$
(B) $\frac{3^{10}}{2^{20}}$
(C) $\frac{3^{20}}{2^{40}}$
(D) $\frac{13 \cdot 17 \cdot 19 \cdot 31}{2^{18}}$
(E) $\frac{3 \cdot 7 \cdot 13 \cdot 17 \cdot 19}{2^{18}}$