since the events are independent, just multiply their probabilities. For the first, that would be
1/6 * 1/2
and the others likewise
-Rolling a 4 then an even number
-Rolling a 3 then a 3
-Rolling a number less than 1 and then a number less than 2
1/6 * 1/2
and the others likewise
Let's start by calculating the sample space (total number of possible outcomes) for each scenario:
1. Rolling a number cube twice:
The number cube has 6 faces, so for each roll, there are 6 possible outcomes. Since you roll it twice, the total number of possible outcomes is 6 * 6 = 36.
Now let's calculate the probability for each event:
1. Rolling a 4 then an even number:
To roll a 4 on the first attempt, there is only 1 successful outcome (rolling a 4). On the second attempt, there are 3 successful outcomes (rolling a 2, 4, or 6). Therefore, the number of successful outcomes is 1 * 3 = 3.
So the probability of rolling a 4 then an even number is 3/36.
2. Rolling a 3 then a 3:
To roll a 3 on both attempts, there is only 1 successful outcome for each roll. Therefore, the number of successful outcomes is 1 * 1 = 1.
So the probability of rolling a 3 then a 3 is 1/36.
3. Rolling a number less than 1 and then a number less than 2:
Since there is no face on the number cube that is less than 1, it is impossible to roll a number less than 1. Therefore, the probability of rolling a number less than 1, followed by any other number, is 0/36, which simplifies to 0.
To summarize:
1. Probability of rolling a 4 then an even number: 3/36
2. Probability of rolling a 3 then a 3: 1/36
3. Probability of rolling a number less than 1 and then a number less than 2: 0/36 (which is 0)