Let's find the probabilities of various outcomes when two dice are rolled together. When rolling two six-sided dice, there are a total of 6 * 6 = 36 possible outcomes.
Given:
- Total possible outcomes = 36
Let's go through each part of the question in detail:
### (i) Sum of digits to be more than 6
To find the probability that the sum of the digits (numbers on the dice) is more than 6, we look at the number of successful outcomes where this condition is met.
- Number of successful outcomes = 21
The probability is:
[tex]\[ \text{Probability} = \frac{\text{Number of successful outcomes}}{\text{Total possible outcomes}} = \frac{21}{36} = 0.5833 \][/tex]
### (ii) Sum of digits to be less than 3
To find the probability that the sum of the digits is less than 3, we consider the relevant outcomes:
- Number of successful outcomes = 1
The probability is:
[tex]\[ \text{Probability} = \frac{\text{Number of successful outcomes}}{\text{Total possible outcomes}} = \frac{1}{36} = 0.0278 \][/tex]
### (iii) Sum of digits to be either 5 or 6
To find the probability that the sum of the digits is either 5 or 6, we consider the number of outcomes where this condition is met:
- Number of successful outcomes = 9
The probability is:
[tex]\[ \text{Probability} = \frac{\text{Number of successful outcomes}}{\text{Total possible outcomes}} = \frac{9}{36} = 0.25 \][/tex]
### (iv) Sum of digits to be 12
To find the probability that the sum of the digits is exactly 12, we consider the relevant outcomes:
- Number of successful outcomes = 1
The probability is:
[tex]\[ \text{Probability} = \frac{\text{Number of successful outcomes}}{\text{Total possible outcomes}} = \frac{1}{36} = 0.0278 \][/tex]
### (v) Sum of digits to be less than 12 but more than 5
To find the probability that the sum of the digits falls in the range between 5 (exclusive) and 12 (exclusive), we count those outcomes:
- Number of successful outcomes = 25
The probability is:
[tex]\[ \text{Probability} = \frac{\text{Number of successful outcomes}}{\text{Total possible outcomes}} = \frac{25}{36} = 0.6944 \][/tex]
Thus, summarizing the probabilities:
1. Probability of sum more than 6: [tex]\( 0.5833 \)[/tex]
2. Probability of sum less than 3: [tex]\( 0.0278 \)[/tex]
3. Probability of sum either 5 or 6: [tex]\( 0.25 \)[/tex]
4. Probability of sum equal to 12: [tex]\( 0.0278 \)[/tex]
5. Probability of sum less than 12 but more than 5: [tex]\( 0.6944 \)[/tex]
