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.
answer please :) i don't get this question
If one shows a 6, to have a sum ≥ 10, the other die must show a 4, 5 or 6.
I'm not sure which probability you are seeking. If it is just the black die, then probability = 3/6 = 1/2
If for both events, P(red) for 6 = 1/6. The probability of both/all events occurring is found by multiplying the probabilities of the individual events.
The events "the red cube shows a 6" and "the sum is greater than or equal to 10" are dependent because the outcome of one event (the red cube showing a 6) can affect the outcome of the other event (the sum being greater than or equal to 10).
To find the probability, let's first determine the possible outcomes of rolling the two number cubes. Each cube has six faces, numbered 1 to 6. So, when two number cubes are rolled, there are a total of 6 * 6 = 36 possible outcomes.
We want to find the probability of the event "the red cube shows a 6" and "the sum is greater than or equal to 10". Let's break it down into two scenarios:
1. The red cube shows a 6 and the black cube shows a number between 4 and 6 (inclusive): In this scenario, there are 3 possible outcomes (6, 4), (6, 5), and (6, 6). So, the probability of this scenario is 3/36 or 1/12.
2. The red cube shows a 6 and the black cube shows a number between 1 and 3: In this scenario, there are 3 possible outcomes (6, 1), (6, 2), and (6, 3). So, the probability of this scenario is also 3/36 or 1/12.
Now, we add the probabilities of the two scenarios to find the overall probability:
P("the red cube shows a 6" and "the sum is greater than or equal to 10") = P("the red cube shows a 6" and "the black cube shows a number between 4 and 6") + P("the red cube shows a 6" and "the black cube shows a number between 1 and 3")
= 1/12 + 1/12
= 2/12
= 1/6
Therefore, the probability of the events "the red cube shows a 6" and "the sum is greater than or equal to 10" is 1/6.
No problem, I'll explain it to you step by step.
When we talk about events being dependent, it means that the outcome of one event affects the outcome of the other event. In this case, the events are "the red cube shows a 6" and "the sum is greater than or equal to 10".
To understand why they are dependent, let's consider the possible outcomes of rolling both number cubes. Assume that the red cube has numbers from 1 to 6 and the black cube also has numbers from 1 to 6.
Now, if we focus on the event "the red cube shows a 6", we have six possible outcomes since the red cube has six sides. Out of these six outcomes, only one outcome satisfies the condition of rolling a 6 on the red cube.
Next, let's consider the event "the sum is greater than or equal to 10". To find the total number of outcomes where the sum is greater than or equal to 10, we need to analyze all the possible combinations. Here are the combinations that satisfy this condition:
- (4, 6)
- (5, 5)
- (5, 6)
- (6, 4)
- (6, 5)
- (6, 6)
As we can see, all these combinations have a 6 on the red cube. Therefore, the outcome of the event "the red cube shows a 6" affects the outcome of the event "the sum is greater than or equal to 10". This demonstrates the dependency between the two events.
Now, let's calculate the probability of these dependent events occurring. To find the probability, we need to divide the number of favorable outcomes by the number of possible outcomes.
Since there is only one favorable outcome for the event "the red cube shows a 6", and the total number of possible outcomes on the red cube is 6, the probability of the red cube showing a 6 is 1/6.
Similarly, there are 6 favorable outcomes for the event "the sum is greater than or equal to 10", and the total number of possible outcomes when rolling two number cubes is 36 (6 x 6). Therefore, the probability of the sum being greater than or equal to 10 is 6/36, which simplifies to 1/6.
Therefore, the probability of both events occurring together (the red cube showing a 6 and the sum being greater than or equal to 10) is the product of their individual probabilities, which is (1/6) * (1/6) = 1/36.
So, the probability of both events happening at the same time is 1/36.