In a gambling between Mr. A and Mr. B a machine continues tossing a fair coin until either HT or TT on consecutive throws are obtained for the first time. If it is HT, Mr. A wins and if it is TT, Mr. B wins, which of the following are true?

$P\left(Awins\right)=P$ (A wins by tossing tail first)+ P(A wins  by tossing head first)

$$\begin{gathered} =P(THU\cup THHT\cup THIHHT\cup \mathrm{.....})+P(THT\cup HHT\cup THHIHT\cup \mathrm{...})+ \\ (\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\mathrm{.....}\infty )+(\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\mathrm{....}\infty )=\frac{1}{4}+\frac{1}{2}=\frac{3}{4} \end{gathered}$$

$P(A$ wins if first toss is head)  $=P\left(TUHTUHHTU\mathrm{....}\right)=\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\mathrm{....}\infty =1$

P(A Wins  if first toss is tall)   $=P\left(HTUHHTUHHHTU\mathrm{......}\right)=\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\mathrm{.....}\infty =\frac{1}{2}$

$NowP\left(A\right):P\left(B\right)=30:31$