Recognising a binomial model
A binomial model counts how many times one chosen outcome occurs in a fixed number of trials. The formula is useful only after the model has passed four checks.
Decide what one trial and one success mean
A trial is one repeat of the experiment. A success is the outcome being counted; it does not have to be good. If counts defective items, then “defective” is called a success.
- = the fixed number of trials.
- = the probability of success on one trial.
- = the probability of failure on one trial.
- = the number of successes in all trials.
Symbols
- = number of successes (none)
- = fixed number of trials (none)
- = success probability on each trial (none)
The symbol means “has the distribution”. Because the count can range from no successes to success every time,
A binomial model must pass four checks
- Fixed trials. The number is known before the experiment starts.
- Two categories. Every trial is classified as success or failure.
- Independent trials. One result does not change the probability on a later trial.
- Constant probability. The same applies on every trial.
Use the same success/failure trials. Change what the experiment records.
Count successes in all 5 trials
1
2
3
4
5
X = 2 successes
possible values: 0, 1, 2, 3, 4, 5
Reject the model when a condition fails
Worked example
Why sampling without replacement can fail
Three cards are drawn without replacement from a small pack. Even if each draw is labelled “ace” or “not ace”, removing a card changes the probabilities. The trials are not independent and is not constant, so an exact binomial model is not suitable.
Common mistake
A spinner lands red with probability 0.3. It is spun independently 12 times. Let be the number of reds. State the distribution of , its possible values and .
Show worked answer
There are 12 fixed independent trials, each has two categories, and the red probability stays 0.3.