A salesperson finds that, in the long run, two out of five sales calls are successful. Twelve calls are to be made. Let X = the number of concluded sales. Is X a binomial random variable? Explain!
Yes. With a binomial distribution, you have two outcomes. In this case, the outcomes are success or failure.
To determine if X is a binomial random variable, we need to check if it satisfies the characteristics of a binomial distribution:
1. Fixed number of trials: In this case, the salesperson is making twelve calls. So, the number of trials is fixed.
2. Independent trials: Each sales call is treated as an independent event, assuming that the outcome of one call does not affect the outcome of another call.
3. Two possible outcomes: Each call can either be successful or unsuccessful.
4. Constant probability of success: The problem states that, in the long run, two out of five sales calls are successful. This indicates a constant probability of success for each call.
If all these conditions are met, then X can be considered a binomial random variable.
In this scenario, we know that the number of trials is fixed (12), each call is treated as an independent event, there are two possible outcomes (successful or unsuccessful), and the probability of success remains constant. Therefore, X can be considered a binomial random variable.
To determine whether X is a binomial random variable, we need to check if it satisfies the criteria for a binomial distribution.
The criteria for a binomial distribution are as follows:
1. The experiment consists of a fixed number of trials.
2. Each trial can only result in two outcomes: success or failure.
3. The probability of success is constant for each trial.
4. The trials are independent of each other.
In this case, we have 12 sales calls to be made, which satisfies the first criteria: a fixed number of trials.
The second criteria require that each trial results in either a success or failure. Here, we are told that two out of five sales calls are successful, so we can say that for each trial, success is defined as making a sale and failure is defined as not making a sale.
The third criteria state that the probability of success is constant for each trial. In this case, the probability of success is 2/5, which remains the same for each sales call.
Finally, the fourth criteria state that the trials should be independent of each other. It means that the outcome of one trial should not affect the outcome of another trial. While this assumption might not hold perfectly in real-world scenarios, we can assume independence for the purpose of this question.
Therefore, based on these criteria, we can conclude that X, the number of concluded sales, is indeed a binomial random variable in this scenario.