Why does the binomial distribution require independent trials with a constant success probability?

Conditions

• Each trial has exactly two mutually exclusive outcomes (success or failure).
• Trials are repeated independently.
• The probability of success remains constant across all trials.

Reasoning, step by step

1. Identify the probability of a specific sequence of k successes and n-k failures as $p^k(1-p)^{n-k}$.
2. Recognize that this multiplication assumes the outcome of one trial does not affect another.
3. Note that the binomial coefficient $\binom{n}{k}$ counts all possible arrangements of these successes.
4. Conclude that without independence and constant probability, the probability of any specific sequence is not simply the product of individual probabilities.

Example

In an NBA season, if a team plays back-to-back games, player fatigue and coach rotations can make the second game's win probability depend on the first game. The video states: 'Fatigue and rotation in back-to-back games can create dependence or change the win probability... These examples show why a binomial model should not be applied automatically to a real season.'

Common misconceptions

• Assuming any repeated binary experiment automatically follows a binomial distribution.
• Believing that the binomial formula can be used even if the success probability fluctuates from trial to trial.

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Connected concepts

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• How does the binomial probability mass function calculate the probability of exactly k successes?
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