
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
Onboard an airliner, there are $2n$ passengers, and the airline has provisioned $n$ portions of chicken and $n$ portions of fish. Each passenger prefers chicken with a probability of 0.5 and fish with a probability of 0.5. We call a passenger unhappy if they end up with the meal that they do not prefer.

a) Find the most probable number of unhappy passengers.

b) Find the expected number of unhappy passengers.

c) Find the variance of the number of unhappy passengers.

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