In which distributions does the variable, X, have a binomial distribution?
(Choose all that are correct)
~When a 6-sided number cube is rolled 3 times, X is the number of 2s rolled.
~When a 6-sided number cube is rolled, X is the number of times rolled before a 2 appears.
~When a 6-sided number cube is rolled twice, X is the number of odd numbers rolled.
~When a 6-sided number cube is rolled twice, X is the sum of the numbers rolled.
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a number cube with the numbers one through six is rolled and a spinner with the colors blue, yellow, another yellow, and green is spun what is the probability of rolling an odd number and yellow
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A number cube is rolled 360 times, and the results are recorded as follows: 96 ones, 31 twos, 49 threes, 76 fours, and 63 sixes. What is the experimental probability of rolling a 2 or 3
Hi! The total number of times the number cube was rolled is 360, and the number of times a 2 or 3 turned up is 31+49=80. So, the experimental probability of rolling a 2 or 3 is 80/360, which simplifies to 4/18 or 2/9. Therefore, the experimental probability of rolling a 2 or 3 is approximately 0.222 or 22.2%, which is approximately 2 chances in 9.
To determine if a variable follows a binomial distribution, we need to check if it satisfies the following conditions:
1. The variable must have a fixed number of trials or experiments.
2. Each trial must have only two possible outcomes, typically referred to as success and failure.
3. The probability of success must remain constant across all trials.
4. The trials must be independent, meaning that the outcome of one trial does not affect the outcome of another trial.
Let's analyze each scenario:
1. When a 6-sided number cube is rolled 3 times, X is the number of 2s rolled.
In this case, X follows a binomial distribution because:
- We have a fixed number of trials (3 rolls of the number cube).
- Each trial has two possible outcomes (getting a 2 or not getting a 2).
- The probability of success (getting a 2) remains constant across all trials.
- The trials are independent.
2. When a 6-sided number cube is rolled, X is the number of times rolled before a 2 appears.
In this case, X does not follow a binomial distribution because:
- The number of trials is not fixed since we don't know how many times we need to roll before a 2 appears.
- The outcomes are not simply success and failure, but rather the number of rolls needed.
- The probability of success changes with each roll, as the probability of obtaining a 2 increases with every attempt.
- The trials are not independent, as the outcome of one roll affects the probability of success in the following roll.
3. When a 6-sided number cube is rolled twice, X is the number of odd numbers rolled.
In this case, X follows a binomial distribution because:
- We have a fixed number of trials (2 rolls of the number cube).
- Each trial has two possible outcomes (getting an odd number or not getting an odd number).
- The probability of success (getting an odd number) remains constant across all trials.
- The trials are independent.
4. When a 6-sided number cube is rolled twice, X is the sum of the numbers rolled.
In this case, X does not follow a binomial distribution because:
- The number of trials is fixed (2 rolls), but the outcome is not a count or discrete value like the number of successes.
- The number of possible outcomes is not limited to two, as any sum of two dice values from 2 to 12 is possible.
- The probability of success is not fixed across all trials, as different sums of the two rolls have different probabilities.
- The trials are independent, but this alone does not meet all the criteria for a binomial distribution.
Therefore, the distributions where the variable X follows a binomial distribution are:
- When a 6-sided number cube is rolled 3 times, X is the number of 2s rolled.
- When a 6-sided number cube is rolled twice, X is the number of odd numbers rolled.