A scratch-off lottery game with 12 boxes labeled 1 to 12. Each game involves tossing a fair coin four times to determine which boxes to scratch. The rules are as follows: (I) On the first coin toss, if heads, scratch box 1; if tails, scratch box 3. (II) On the second, third, and fourth coin tosses, if heads, the number of the box to scratch is the number of the previous box plus 1; if tails, the number of the box to scratch is the number of the previous box plus 3, and so on. Example: If the results of four coin tosses are ``heads, tails, tails, heads'' in order, then boxes numbered 1, 4, 7, and 8 will be scratched. Let $p _ { m }$ denote the probability that box $m$ is scratched in each game. Select the correct options. (1) $p _ { 2 } = \frac { 1 } { 4 }$ (2) $p _ { 3 } = \frac { 1 } { 2 }$ (3) $p _ { 4 } = \frac { 1 } { 2 } p _ { 1 } + \frac { 1 } { 2 } p _ { 3 }$ (4) $p _ { 8 } > p _ { 10 }$ (5) Given that box 4 is scratched, the probability that box 3 is scratched is $\frac { 1 } { 2 }$
A scratch-off lottery game with 12 boxes labeled 1 to 12. Each game involves tossing a fair coin four times to determine which boxes to scratch. The rules are as follows:\\
(I) On the first coin toss, if heads, scratch box 1; if tails, scratch box 3.\\
(II) On the second, third, and fourth coin tosses, if heads, the number of the box to scratch is the number of the previous box plus 1; if tails, the number of the box to scratch is the number of the previous box plus 3, and so on.\\
Example: If the results of four coin tosses are ``heads, tails, tails, heads'' in order, then boxes numbered 1, 4, 7, and 8 will be scratched.\\
Let $p _ { m }$ denote the probability that box $m$ is scratched in each game. Select the correct options.\\
(1) $p _ { 2 } = \frac { 1 } { 4 }$\\
(2) $p _ { 3 } = \frac { 1 } { 2 }$\\
(3) $p _ { 4 } = \frac { 1 } { 2 } p _ { 1 } + \frac { 1 } { 2 } p _ { 3 }$\\
(4) $p _ { 8 } > p _ { 10 }$\\
(5) Given that box 4 is scratched, the probability that box 3 is scratched is $\frac { 1 } { 2 }$