What discrete probability distributions result from counting successes in a fixed block of trials versus counting failures before a fixed number of successes?
Conditions
- Trials are independent with a fixed success probability.
- The first scenario involves a fixed total number of trials.
- The second scenario involves waiting for a fixed number of successes.
Reasoning, step by step
- Determine if the total number of trials is fixed or if the number of successes is fixed.
- If the total number of trials is fixed, count the successes to get a binomial variable.
- If the number of successes is fixed, count the failures before reaching that number to get a negative binomial variable.
Example
The script states: 'Counting successes in a fixed block of trials gives a binomial variable. Counting failures before a fixed number of successes gives a negative binomial variable.'
Common misconceptions
- Confusing the binomial distribution (fixed trials, random successes) with the negative binomial distribution (fixed successes, random trials).
- Assuming the negative binomial distribution counts the total number of trials rather than the number of failures.
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