two standard fair 6 sided dice are rolled. What is the probability that their values will differ by exactly two
So we would be looking at the following cases:
1-3, 2-4, 3-5, 4-6 or 3-1, 4-2, 5-3, 6-4,
that is, there are 8 such cases
prob(of your event) = 8/36 = 2/9
6-4
5-3
4-2
3-1
4-6
3-5
2-4
1-3 so 8 work out of how many
6 * 6 I think or 36
so I get
8/36 = 2/9
see:
http://www.freemathhelp.com/rolling-dice.html
To find the probability that the values of two standard fair 6-sided dice differ by exactly two, we need to count the number of favorable outcomes and divide it by the total number of possible outcomes.
Step 1: Understand the problem:
- We have two standard fair 6-sided dice.
- We need to find the probability of the difference between their values being exactly two.
Step 2: Determine the favorable outcomes:
- To count the favorable outcomes, we can consider the two possible situations:
1. The first die shows a number from 1 to 4, and the second die shows a number two greater.
- Favorable outcomes: (1, 3), (2, 4), (3, 5), (4, 6) -> Total of 4 outcomes
2. The first die shows a number from 2 to 5, and the second die shows a number two smaller.
- Favorable outcomes: (2, 4), (3, 5), (4, 6), (5, 1) -> Total of 4 outcomes
Step 3: Determine the total number of outcomes:
- Since each die has 6 possible outcomes, the total number of outcomes for two dice is 6 * 6 = 36.
Step 4: Calculate the probability:
- The probability is the ratio of favorable outcomes to the total number of outcomes.
- Probability = (Number of favorable outcomes) / (Total number of outcomes)
- Probability = (4 + 4) / 36
- Probability = 8 / 36
- Simplified, Probability = 2 / 9
Step 5: Answer the question:
- The probability that the values of two standard fair 6-sided dice differ by exactly two is 2/9.
To find the probability that the values of two standard fair 6-sided dice will differ by exactly two, we need to determine the number of favorable outcomes (values that differ by exactly two) and the total number of possible outcomes.
1. Determine the favorable outcomes:
- We can consider the two dice independently. To have a difference of exactly two, we can look at the possibilities for the higher number: 2, 3, 4, or 5 (since the lower number will be two less).
- For each of these possibilities, we need to determine the number of ways to roll the dice to get those values. For example, if the higher number is 3, the possible combinations are (1, 3) or (3, 1). We have two possible combinations for each higher number.
- So, the total number of favorable outcomes is 2 possibilities for each of the 2, 3, 4, and 5 higher numbers, which gives us 2 * 4 = 8 favorable outcomes.
2. Determine the total number of possible outcomes:
- Since each die has 6 equally likely outcomes, the total number of possible outcomes for two dice is 6 * 6 = 36.
3. Calculate the probability:
- Probability is given by dividing the favorable outcomes by the total number of possible outcomes.
- Therefore, the probability of the dice values differing by exactly two is 8 favorable outcomes divided by 36 total outcomes, which simplifies to 2/9 or approximately 0.2222.
So, the probability that the values of two standard fair 6-sided dice will differ by exactly two is 2/9 or approximately 0.2222.