If $$E$$ and $$F$$ are independent events such that $$0 < P\left( E \right) < 1$$ and $$0 < P\left( F \right) < 1,$$ then

Suppose the probability for A to win a game against B is $$0.4.$$ If $$A$$ has an option of playing either a "best of $$3$$ games" or a "best of $$5$$ games" match against $$B$$, which option should be choose so that the probability of his winning the match is higher ? (No game ends in a draw).

For any three vectors $${\overrightarrow a ,\,\overrightarrow b ,}$$ and $${\overrightarrow c ,}$$
$$\left( {\overrightarrow a - \overrightarrow b } \right)\,.\,\left( {\overrightarrow b - \overrightarrow c } \right)\, \times \,\left( {\overrightarrow c - \overrightarrow a } \right)\, = \,2\overrightarrow {a\,} .\,\overrightarrow {b\,} \times \,\overrightarrow c .$$

If vectors $$\overrightarrow A ,\overrightarrow B ,\overrightarrow C $$ are coplanar, show that $$$\left| {\matrix{ {} & {\overrightarrow {a.} } & {} & {\overrightarrow {b.} } & {} & {\overrightarrow {c.} } \cr {\overrightarrow {a.} } & {\overrightarrow {a.} } & {\overrightarrow {a.} } & {\overrightarrow {b.} } & {\overrightarrow {a.} } & {\overrightarrow {c.} } \cr {\overrightarrow {b.} } & {\overrightarrow {a.} } & {\overrightarrow {b.} } & {\overrightarrow {b.} } & {\overrightarrow {b.} } & {\overrightarrow {c.} } \cr } } \right| = \overrightarrow 0 $$$