Two number cubes are rolled—one red and one black. Explain why the events
“the red cube shows a 6” and “the sum is greater than or equal to 10” are dependent,
and find the probability.
The events "the red cube shows a 6" and "the sum is greater than or equal to 10" are dependent because the outcome of the red cube showing a 6 affects the outcome of the sum being greater than or equal to 10.
To find the probability, we need to first determine the sample space, which is all possible outcomes when rolling two number cubes. Each cube has 6 possible outcomes (numbers 1 to 6), so the total number of outcomes is 6*6 = 36.
To calculate the probability of the first event, the red cube showing a 6, we consider that there is only one outcome out of the 36 possible outcomes where the red cube shows a 6.
Next, we calculate the probability of the second event, the sum being greater than or equal to 10. We need to determine the number of favorable outcomes where the sum is greater than or equal to 10. These outcomes are (4, 6), (5, 6), (6, 4), (6, 5), and (6, 6), which amounts to 5 outcomes out of the 36.
Therefore, the probability of the red cube showing a 6 and the sum being greater than or equal to 10 is 1/36 * 5/36 = 5/1296.
The events "the red cube shows a 6" and "the sum is greater than or equal to 10" are dependent because they are influenced by each other's outcome. In order for the sum of the two numbers to be greater than or equal to 10, the red cube must show a 6. If the red cube shows any number less than 6, it is not possible for the sum to be 10 or greater.
To find the probability, let's first determine the number of favorable outcomes and the total number of possible outcomes.
The red cube has 6 possible outcomes (numbers 1 to 6), and since we're interested in the event that it shows a 6, there is only 1 favorable outcome.
The black cube also has 6 possible outcomes (numbers 1 to 6), and for the sum to be greater than or equal to 10, the black cube must show either a 4, 5, or 6. Thus, there are 3 favorable outcomes.
To calculate the probability, we divide the number of favorable outcomes by the total number of possible outcomes. In this case, the probability is calculated as:
Probability = (Number of favorable outcomes) / (Total number of possible outcomes)
= 1 / 6 * 3 / 6
= 1/36
Therefore, the probability that the red cube shows a 6 and the sum is greater than or equal to 10 is 1/36.