
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
In the game of Rock Paper Scissors, typically two opponents make, with their hand, either a rock, a piece of paper, or a pair of scissors. If the two players play rock and scissors, the one who plays rock wins. If they play scissors and paper, the one who plays scissors wins, and if they play paper and rock, the one who plays paper wins. If they play the same thing, the result is a tie.**p11.** Al and Bob are playing Rock Paper Scissors. Al plays rock. What is the probability that Al wins, given that Bob plays randomly and has an equal probability of playing rock, paper, and scissors?**p12.** In a given game, what is the probability of a tie, given that both players play randomly and with an equal probability of playing rock, paper, and scissors?**p13.** Al and Bob play Rock Paper Scissors until someone wins a game. What is the probability that this happens on the sixth game?**p14.** Al and Bob are joined by Carl and D’Angelo, and they decide to play a team game of Rock Paper Scissors. A game is called perfect if some two of the four play the same thing, and the other two also play the same thing, but something different. For example, an example of a perfect game would be Al and Bob playing rock, and Carl and D’Angelo playing scissors, but if all four play paper, we do not have a perfect game. What is the probability of a perfect game?**p15.** Al is bored of Rock Paper Scissors, and wants to invent a new game:  $Z-Y-X-W-V$ . Two players, each choose to play either  $Z, Y, X, W$ , or  $V$ . If they play the same thing, the result is a tie. However, Al must come up with a ’pecking order’, that is, he must decide which plays beat which. For each of the  $10$  pairs of distinct plays that the two players can make, Al randomly decides a winner. For example, he could decide that  $W$  beats  $Y$  and that  $Z$  beats  $X$ , etc. What is the probability that after Al makes all of these  $10$  choices, the game is balanced, that is, playing each letter results in an equal probability of winning?


PS. You should use hide for answers.

Present the answer in LaTex format: \boxed{Your answer}
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