
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

On board an airliner, there are $2n$ passengers, and the airline has loaded $n$ chicken meals and $n$ fish meals for them. It is known that a passenger prefers chicken with a probability of 0.5, and fish with a probability of 0.5. We will say a passenger is dissatisfied if they do not get what they prefer.

a) (8th - 11th grade, 2 points) Find the most probable number of dissatisfied passengers.

b) (9th - 11th grade, 3 points) Find the expected number of dissatisfied passengers.

c) (9th - 11th grade, 4 points) Find the variance of the number of dissatisfied passengers.

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