Answer :
To solve the probabilities for the events related to rolling a die with sides numbered 1 to 6, we will determine the number of favorable outcomes divided by the total number of possible outcomes.
1. Let's start with determining the probability of rolling a 5:
[tex]\[ P(5) = \frac{\text{Number of outcomes that are 5}}{\text{Total number of outcomes}} \][/tex]
Since there is only one face with the number 5, and there are six sides in total:
[tex]\[ P(5) = \frac{1}{6} \approx 0.1667 \][/tex]
2. Now, let's find the probability of rolling a 1 or a 2:
[tex]\[ P(1 \text{ or } 2) = \frac{\text{Number of outcomes that are 1 or 2}}{\text{Total number of outcomes}} \][/tex]
Since there are two favorable outcomes (rolling a 1 or a 2):
[tex]\[ P(1 \text{ or } 2) = \frac{2}{6} = \frac{1}{3} \approx 0.3333 \][/tex]
3. Next, we calculate the probability of rolling an odd number:
[tex]\[ P(\text{odd number}) = \frac{\text{Number of odd number outcomes}}{\text{Total number of outcomes}} \][/tex]
The odd numbers on the die are 1, 3, and 5, which are three outcomes:
[tex]\[ P(\text{odd number}) = \frac{3}{6} = \frac{1}{2} = 0.5 \][/tex]
4. For the probability of rolling any number that is not a 6:
[tex]\[ P(\text{not 6}) = \frac{\text{Number of outcomes that are not 6}}{\text{Total number of outcomes}} \][/tex]
Since there are 5 numbers that are not 6 (i.e., 1, 2, 3, 4, 5):
[tex]\[ P(\text{not 6}) = \frac{5}{6} \approx 0.8333 \][/tex]
5. Let's determine the probability of rolling an even number:
[tex]\[ P(\text{even number}) = \frac{\text{Number of even number outcomes}}{\text{Total number of outcomes}} \][/tex]
The even numbers on the die are 2, 4, and 6, which are three outcomes:
[tex]\[ P(\text{even number}) = \frac{3}{6} = \frac{1}{2} = 0.5 \][/tex]
6. Finally, the probability of rolling a number that is either 1, 2, 3, or 4:
[tex]\[ P(1, 2, 3 \text{ or } 4) = \frac{\text{Number of outcomes that are 1, 2, 3 or 4}}{\text{Total number of outcomes}} \][/tex]
Since there are four outcomes (1, 2, 3, 4):
[tex]\[ P(1, 2, 3 \text{ or } 4) = \frac{4}{6} = \frac{2}{3} \approx 0.6667 \][/tex]
By summarizing our results, the probabilities are as follows:
5. [tex]\( P(5) = 0.1667 \)[/tex]
6. [tex]\( P(1 \text{ or } 2) = 0.3333 \)[/tex]
7. [tex]\( P(\text{odd number}) = 0.5 \)[/tex]
8. [tex]\( P(\text{not 6}) = 0.8333 \)[/tex]
9. [tex]\( P(\text{even number}) = 0.5 \)[/tex]
10. [tex]\( P(1, 2, 3 \text{ or } 4) = 0.6667 \)[/tex]
1. Let's start with determining the probability of rolling a 5:
[tex]\[ P(5) = \frac{\text{Number of outcomes that are 5}}{\text{Total number of outcomes}} \][/tex]
Since there is only one face with the number 5, and there are six sides in total:
[tex]\[ P(5) = \frac{1}{6} \approx 0.1667 \][/tex]
2. Now, let's find the probability of rolling a 1 or a 2:
[tex]\[ P(1 \text{ or } 2) = \frac{\text{Number of outcomes that are 1 or 2}}{\text{Total number of outcomes}} \][/tex]
Since there are two favorable outcomes (rolling a 1 or a 2):
[tex]\[ P(1 \text{ or } 2) = \frac{2}{6} = \frac{1}{3} \approx 0.3333 \][/tex]
3. Next, we calculate the probability of rolling an odd number:
[tex]\[ P(\text{odd number}) = \frac{\text{Number of odd number outcomes}}{\text{Total number of outcomes}} \][/tex]
The odd numbers on the die are 1, 3, and 5, which are three outcomes:
[tex]\[ P(\text{odd number}) = \frac{3}{6} = \frac{1}{2} = 0.5 \][/tex]
4. For the probability of rolling any number that is not a 6:
[tex]\[ P(\text{not 6}) = \frac{\text{Number of outcomes that are not 6}}{\text{Total number of outcomes}} \][/tex]
Since there are 5 numbers that are not 6 (i.e., 1, 2, 3, 4, 5):
[tex]\[ P(\text{not 6}) = \frac{5}{6} \approx 0.8333 \][/tex]
5. Let's determine the probability of rolling an even number:
[tex]\[ P(\text{even number}) = \frac{\text{Number of even number outcomes}}{\text{Total number of outcomes}} \][/tex]
The even numbers on the die are 2, 4, and 6, which are three outcomes:
[tex]\[ P(\text{even number}) = \frac{3}{6} = \frac{1}{2} = 0.5 \][/tex]
6. Finally, the probability of rolling a number that is either 1, 2, 3, or 4:
[tex]\[ P(1, 2, 3 \text{ or } 4) = \frac{\text{Number of outcomes that are 1, 2, 3 or 4}}{\text{Total number of outcomes}} \][/tex]
Since there are four outcomes (1, 2, 3, 4):
[tex]\[ P(1, 2, 3 \text{ or } 4) = \frac{4}{6} = \frac{2}{3} \approx 0.6667 \][/tex]
By summarizing our results, the probabilities are as follows:
5. [tex]\( P(5) = 0.1667 \)[/tex]
6. [tex]\( P(1 \text{ or } 2) = 0.3333 \)[/tex]
7. [tex]\( P(\text{odd number}) = 0.5 \)[/tex]
8. [tex]\( P(\text{not 6}) = 0.8333 \)[/tex]
9. [tex]\( P(\text{even number}) = 0.5 \)[/tex]
10. [tex]\( P(1, 2, 3 \text{ or } 4) = 0.6667 \)[/tex]
