A red die and a green die have been rolled. (6 sided) What is the probability of the event?
1.The number on the red die is greater than the number on the green die (Answer is 5/12 but how?)
2. The sum is less than 10.
3. The sum is even.
4. The sum is prime.
5. The sum is 7 or 11.
draw a grid (matrix) of the 2 dice,
RED across the top
GREEN along the column
You will that for 6 of them , they are equal, e.g. 3,3
For 15 of them the green is more
fro 15 of them the red is more , (6+15+15=36)
so prob(red>green) = 15/36 = 5/12
from your chart, count how many have a sum < 10
easy way: count those 10 or more:
6,6 ; 6,5 ; 6,4 ; 5,6 ; 4,6 ; 5,5
i count 6, check that. Did I list all of them?
So less than 10 = 36-6 = 30
prob(less than 10) = (36-6)/36 = 5/6
do the rest of the problems with the help of your chart
To calculate the probabilities of these events, we need to determine the total number of possible outcomes and the number of favorable outcomes for each event. In the case of rolling two six-sided dice, we have 36 possible outcomes (since each die has 6 sides).
1. The number on the red die is greater than the number on the green die:
We can find the number of favorable outcomes by counting the pairs of numbers where the red die is greater than the green die. These pairs are: (2,1), (3,1), (4,1), (5,1), (6,1), (3,2), (4,2), (5,2), (6,2), (4,3), (5,3), (6,3), (5,4), (6,4), and (6,5). This gives us a total of 15 favorable outcomes. Therefore, the probability is 15/36, which simplifies to 5/12.
2. The sum is less than 10:
We can find the number of favorable outcomes by counting the pairs of numbers that have a sum less than 10. These pairs are: (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (3,1), (3,2), (3,3), (3,4), (4,1), (4,2), (4,3), (5,1), (5,2), and (6,1). This gives us a total of 21 favorable outcomes. Therefore, the probability is 21/36, which simplifies to 7/12.
3. The sum is even:
To find the number of favorable outcomes, we need to count the pairs of numbers that have an even sum. These include pairs where both numbers are even (1 possible outcome: (2,2)) and pairs where one number is even and the other is odd (15 possible outcomes: (1,2), (1,4), (1,6), (2,1), (2,3), (2,5), (3,2), (3,4), (4,1), (4,3), (5,2), (5,4), (6,1), (6,3), (6,5)). This gives us a total of 16 favorable outcomes. Therefore, the probability is 16/36, which simplifies to 4/9.
4. The sum is prime:
The prime numbers less than or equal to 12 (the maximum sum of two dice) are 2, 3, 5, 7, and 11. We need to count the pairs of numbers that have a sum equal to one of these primes: (1,1), (1,2), (1,4), (1,6), (2,1), (2,3), (2,5), (3,2), (3,4), (4,1), (4,3), (5,2), (5,6), (6,1), (6,5). This gives us a total of 15 favorable outcomes. Therefore, the probability is 15/36, which simplifies to 5/12.
5. The sum is 7 or 11:
We need to count the pairs of numbers that have a sum equal to 7 or 11. These pairs are: (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1). This gives us a total of 6 favorable outcomes. Therefore, the probability is 6/36, which simplifies to 1/6.
To calculate the probability of each event, we need to determine the total number of possible outcomes and the number of favorable outcomes for each event.
1. The number on the red die is greater than the number on the green die:
Total number of outcomes = 6 * 6 = 36 (since each die has 6 sides)
Number of favorable outcomes = 15 (there are 6 outcomes where the red die shows a 2, 4, or 6, and for each of these outcomes, there are 3 possible outcomes on the green die)
Probability = Number of favorable outcomes / Total number of outcomes = 15 / 36 = 5 / 12
2. The sum is less than 10:
Total number of outcomes = 36
Number of favorable outcomes = 27 (excluding the outcomes with sums 10, 11, and 12: (5, 5), (5, 6), (6, 5), (6, 6))
Probability = 27 / 36 = 3 / 4
3. The sum is even:
Total number of outcomes = 36
Number of favorable outcomes = 18 (outcomes with sums 2, 4, 6, 8, 10, and 12)
Probability = 18 / 36 = 1 / 2
4. The sum is prime:
Total number of outcomes = 36
Number of favorable outcomes = 10 (outcomes with sums 2, 3, 5, 7, and 11)
Probability = 10 / 36 = 5 / 18
5. The sum is 7 or 11:
Total number of outcomes = 36
Number of favorable outcomes = 6 (outcomes with sums 7 and 11: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1))
Probability = 6 / 36 = 1 / 6