Two Dice - Probability Sum is 7
You roll two standard fair six-sided dice. What is the probability that the sum of the numbers shown on the two dice equals 7?
Related Concepts
Hint
- First, determine the total number of possible outcomes when rolling two dice. (Think: How many outcomes for die 1? How many for die 2?)
- Next, list all the specific pairs of outcomes (die1, die2) that sum up to 7.
- The probability is the number of these "sum to 7" pairs divided by the total number of possible pairs.
Explanation: Probability Sum is 7
Imagine you have two dice, one red and one blue:
When you roll them, each die can land on a number from 1 to 6. We want to find out how often their sum will be 7.
- All Possible Pairs: The red die has 6 options, and for each of those, the blue die has 6 options. So, there are 6 × 6 = 36 different pairs you can get (like red 1 & blue 1, red 1 & blue 2, ..., red 6 & blue 6).
- Pairs that Sum to 7: Let's list them:
- If red is 1, blue must be 6 (1+6=7)
- If red is 2, blue must be 5 (2+5=7)
- If red is 3, blue must be 4 (3+4=7)
- If red is 4, blue must be 3 (4+3=7)
- If red is 5, blue must be 2 (5+2=7)
- If red is 6, blue must be 1 (6+1=7)
- The Chance: Out of 36 total possible pairs, 6 of them give a sum of 7. So the probability is 6 out of 36.
To find the probability that the sum of two fair six-sided dice equals 7, we follow these steps:
1. Determine the Sample Space (Total Possible Outcomes)
Each die has 6 possible outcomes: {1, 2, 3, 4, 5, 6}.
When rolling two dice, we can think of the outcomes as ordered pairs (Die 1, Die 2). Since the outcome of the first die does not affect the outcome of the second (they are independent), the total number of possible outcomes is the product of the number of outcomes for each die:
Total Outcomes = Outcomes for Die 1 × Outcomes for Die 2 = 6 × 6 = 36
These 36 outcomes are all equally likely because the dice are fair.
2. Identify the Favorable Outcomes (Sum = 7)
We need to list all the pairs of outcomes (d₁, d₂) where d₁ is the result of the first die and d₂ is the result of the second die, such that d₁ + d₂ = 7:
- (1, 6) [1 + 6 = 7]
- (2, 5) [2 + 5 = 7]
- (3, 4) [3 + 4 = 7]
- (4, 3) [4 + 3 = 7]
- (5, 2) [5 + 2 = 7]
- (6, 1) [6 + 1 = 7]
There are 6 favorable outcomes where the sum is 7.
3. Calculate the Probability
The probability of an event (E) is calculated as:
P(E) = (Number of Favorable Outcomes) / (Total Number of Possible Outcomes)
In this case, E is the event that the sum of the two dice is 7.
P(Sum = 7) = 6 / 36
This fraction can be simplified:
P(Sum = 7) = 1/6
As a decimal, this is approximately:
P(Sum = 7) ≈ 0.1667 or 16.67%
Visualizing the Sample Space: It can be helpful to visualize all 36 outcomes in a grid:
| Die 1 \ Die 2 | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| 1 | (1,1) | (1,2) | (1,3) | (1,4) | (1,5) | (1,6) |
| 2 | (2,1) | (2,2) | (2,3) | (2,4) | (2,5) | (2,6) |
| 3 | (3,1) | (3,2) | (3,3) | (3,4) | (3,5) | (3,6) |
| 4 | (4,1) | (4,2) | (4,3) | (4,4) | (4,5) | (4,6) |
| 5 | (5,1) | (5,2) | (5,3) | (5,4) | (5,5) | (5,6) |
| 6 | (6,1) | (6,2) | (6,3) | (6,4) | (6,5) | (6,6) |
The 6 highlighted cells are the outcomes where the sum is 7.
Challenge Yourself: Using the same two dice, what is the probability that the sum is 4? What about a sum greater than 10?