Which of the following are binomial experiments? (more than one is possible)
1.) A store finds that 32% of people who enter the store will make a purchase. During a day, 50 people enter the store. The random variable represents the number of people who don't make a purchase.
2.) A jar contains 5 blue marbles, 6 green marbles, and 3 red marbles. You randomly select 2 marbles from the jar without replacement. You repeat that step. The random variable represents the number of red marbles that you select.
3.) A state lottery randomly chooses 6 balls numbered 1-40. You choose 6 of the 40 numbers and purchase a lottery ticket. The random variable represents the number of matches on your ticket to the numbers drawn in the lottery.
4.) A surgical procedure has a 76% chance of success. A doctor performs the procedure on 12 patients. The random variable represents the number of successful surgeries.
To determine which of the given scenarios are binomial experiments, we need to check if they satisfy the following criteria:
1. The experiment consists of a fixed number of trials.
2. Each trial can only have two possible outcomes, referred to as success and failure.
3. The probability of success remains constant for each trial.
4. The trials are independent of each other.
Now let's analyze each scenario:
1. A store finds that 32% of people who enter the store will make a purchase. During a day, 50 people enter the store. The random variable represents the number of people who don't make a purchase.
In this scenario, we have a fixed number of trials (50 people entering the store), and each trial can only have two outcomes (making a purchase or not). The probability of success (making a purchase) remains constant at 32%. Moreover, we can assume that the trials are independent since one person's decision to make a purchase doesn't affect another person's decision. Therefore, this scenario meets all the criteria for a binomial experiment.
2. A jar contains 5 blue marbles, 6 green marbles, and 3 red marbles. You randomly select two marbles from the jar without replacement. You repeat that step. The random variable represents the number of red marbles that you select.
In this scenario, the number of trials is not fixed since you continue selecting marbles multiple times. Additionally, the probability of success (selecting a red marble) changes with each trial since the number of red marbles decreases after each selection. Therefore, this scenario does not satisfy the criteria for a binomial experiment.
3. A state lottery randomly chooses 6 balls numbered 1-40. You choose 6 out of the 40 numbers and purchase a lottery ticket. The random variable represents the number of matches on your ticket to the numbers drawn in the lottery.
In this scenario, we have a fixed number of trials (selecting 6 numbers) and each trial can only have two outcomes (matching or not). The probability of success (matching a number) remains constant for each trial since the drawn numbers are chosen randomly. The trials are also assumed to be independent since the selection of one number does not affect the selection of another number. Therefore, this scenario satisfies all the criteria for a binomial experiment.
4. A surgical procedure has a 76% chance of success. A doctor performs the procedure on 12 patients. The random variable represents the number of successful surgeries.
In this scenario, we have a fixed number of trials (performing the procedure on 12 patients) and each trial can only have two outcomes (successful surgery or not). The probability of success (successful surgery) remains constant at 76% for each trial. The trials are independent since the success or failure of one surgery does not affect the outcome of another surgery. Therefore, this scenario satisfies all the criteria for a binomial experiment.
Based on the analysis, the binomial experiments are:
1.) A store finds that 32% of people who enter the store will make a purchase. During a day, 50 people enter the store. The random variable represents the number of people who don't make a purchase.
3.) A state lottery randomly chooses 6 balls numbered 1-40. You choose 6 of the 40 numbers and purchase a lottery ticket. The random variable represents the number of matches on your ticket to the numbers drawn in the lottery.
4.) A surgical procedure has a 76% chance of success. A doctor performs the procedure on 12 patients. The random variable represents the number of successful surgeries.
To determine which of the scenarios are binomial experiments, let's consider the criteria for a binomial experiment:
1. The experiment consists of a fixed number of trials.
2. Each trial has only two possible outcomes: success or failure.
3. The probability of success remains the same for each trial.
4. The trials are independent of each other.
Let's evaluate each scenario based on these criteria:
1.) A store finds that 32% of people who enter the store will make a purchase. During a day, 50 people enter the store. The random variable represents the number of people who don't make a purchase.
This scenario meets all the criteria of a binomial experiment. The fixed number of trials is the 50 people entering the store. Each person either makes a purchase (success) or doesn't make a purchase (failure). The probability of success (making a purchase) remains the same at 32%, and the trials are independent of each other.
Therefore, scenario 1 is a binomial experiment.
2.) A jar contains 5 blue marbles, 6 green marbles, and 3 red marbles. You randomly select 2 marbles from the jar without replacement. You repeat that step. The random variable represents the number of red marbles that you select.
This scenario does not meet the criteria for a binomial experiment. The trials are not independent of each other because the marbles are selected without replacement. The probability of success (selecting a red marble) changes with each trial as the number of available red marbles decreases.
Therefore, scenario 2 is not a binomial experiment.
3.) A state lottery randomly chooses 6 balls numbered 1-40. You choose 6 of the 40 numbers and purchase a lottery ticket. The random variable represents the number of matches on your ticket to the numbers drawn in the lottery.
This scenario meets all the criteria of a binomial experiment. The fixed number of trials is 6 balls being chosen. Each ball either matches (success) or doesn't match (failure) the numbers on your ticket. The probability of success (a match) remains the same for each trial (dependent on the number of matching numbers and the total number of possibilities), and the trials are independent of each other.
Therefore, scenario 3 is a binomial experiment.
4.) A surgical procedure has a 76% chance of success. A doctor performs the procedure on 12 patients. The random variable represents the number of successful surgeries.
This scenario meets all the criteria of a binomial experiment. The fixed number of trials is 12 surgeries. Each surgery either succeeds (success) or fails (failure). The probability of success (a successful surgery) remains the same at 76%, and the trials are independent of each other.
Therefore, scenario 4 is a binomial experiment.
In summary, scenarios 1, 3, and 4 are binomial experiments.