You roll two number cubes.
Let event A = You roll an even number on the first cube.
Let event B = You roll a 6 on the second cube.
Are the events independent or dependent? Why?
A. Dependent, because both cubes have six sides.
B. Independent, because the outcome of the first roll doesn't affect the outcome of the second roll.
C. Dependent, because 6 is an even number.
D. Independent, because they have no outcomes in common.
Answer (2)
The events are independent because the outcome of the first roll does not affect the outcome of the second roll. Therefore, the correct answer is B.
Explanation
Analyze the problem When we roll two number cubes, the outcome of the first roll has absolutely no impact on the outcome of the second roll, and vice versa. This is because each cube roll is a separate event.
Determine dependence or independence Since the outcome of the first roll (event A) doesn't change the possible outcomes or probabilities of the second roll (event B), the events are independent.
Select the correct answer Now, let's look at the answer choices: A. Dependent, because both cubes have six sides. - This is incorrect because the number of sides on the cubes doesn't determine dependence. B. Independent, because the outcome of the first roll doesn't affect the outcome of the second roll. - This is the correct answer. C. Dependent, because 6 is an even number. - This is incorrect because the fact that 6 is even doesn't make the events dependent. D. Independent, because they have no outcomes in common. - This is incorrect because even if they had outcomes in common, it wouldn't necessarily make them dependent.
Examples
Understanding independent events is crucial in many real-life scenarios. For instance, consider a game where you flip a coin and then roll a die. The result of the coin flip (heads or tails) does not influence the outcome of the die roll (1 to 6), and vice versa. This concept is also applicable in more complex situations, such as calculating the probability of success in a series of independent trials, like manufacturing processes or scientific experiments, where each trial's outcome doesn't affect the others.
The events A and B are independent because the outcome of the first roll does not influence the outcome of the second roll. Therefore, the correct answer is B.
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