Determine whether the given procedure results in a binomial distribution. If not, state the reason why.
Rolling a single die 33 times, keeping track of the numbers that are rolled.
Answer (1)
The procedure of rolling a single die 33 times does not result in a binomial distribution because each roll has 6 possible outcomes, whereas a binomial distribution requires each trial to have only two possible outcomes.
Explanation
Analyze the problem We are asked to determine if rolling a single die 33 times and keeping track of the numbers rolled results in a binomial distribution. To answer this, we need to understand the properties of a binomial distribution.
List characteristics of binomial distribution A binomial distribution has the following characteristics:
A fixed number of trials.
Each trial is independent.
Each trial has only two possible outcomes: success or failure.
The probability of success is the same for each trial.
Check if the given procedure satisfies the characteristics In this procedure, we roll a die 33 times, so we have a fixed number of trials (33). Each roll is independent of the others. However, each roll of the die has 6 possible outcomes (1, 2, 3, 4, 5, or 6). For a binomial distribution, each trial must have only two possible outcomes.
Conclude Since rolling a die has 6 possible outcomes instead of 2, this procedure does not result in a binomial distribution.
Examples
Consider a quality control process where you inspect 33 items from a production line. If you were only checking whether each item is defective or not (two outcomes), and the probability of an item being defective is constant, then this would be a binomial distribution. However, if you were categorizing each item into one of several defect types, it would not be a binomial distribution.
