If we roll two dice at the same time, the probability that both dice land on the same number (i.e. we roll “doubles”) is 6/36 or 1/6.
To understand why this is the case, we must realize that there are 36 different ways for two dice to land.
For example:
- The first dice might land on 1 and the second dice might land on 1.
- The first dice might land on 1 and the second dice might land on 2.
- The first dice might land on 1 and the second dice might land on 3.
- The first dice might land on 1 and the second dice might land on 4.
- The first dice might land on 1 and the second dice might land on 5.
- The first dice might land on 1 and the second dice might land on 6.
- The first dice might land on 2 and the second dice might land on 1.
- . . .
And so on.
We can create the following grid to visualize each possible combination of outcomes for the two dice:
To find out the probability that both dice land on the same number, we must use the following formula:
Probability = (#Ways to Land on Same Number) / (#Total Ways to Land)
From the grid above, we can see that there are only 6 ways that both dice can land on the same number and there are 36 total possible ways for both dice to land.
Thus, the probability that both dice land on the same number can be calculated as:
- Probability = (#Ways to Land on Same Number) / (#Total Ways to Land)
- Probability = 6 / 36
- Probability = 1/6
The probability that both dice land on the same number is 1/6 or 0.166666666.
Additional Resources
The following tutorials explain other common topics in probability:
3 Dice Probability Chart (With Probabilities)
How to Find Probability of At Least One Head in Coin Flips
How to Find the Probability of Neither A Nor B
How to Find the Probability of A or B
Easier to explain in my opinion. Roll the first die. Whatever number it lands on, there is a 1 on 6 chance of rolling that with the second die. The dice are independent rolls
What are the odds of three people rolling double three’s in a row?
And then rolling a total of eight, in different ways……..just after the double threes? Yes… this actually happened last Friday at mahjong. Totally blew me away…. and I just need to know what the probabilities are. Could you please answer. Thank you.
Hello Zack, Can you help me?
What are the odds of rolling doubles on the first roll followed by the SAME doubles on the next roll with the same die? I think it would be 1/6 X 1/18 but it’s been a very long time since I was in math class.
And then roll the SAME doubles on the third roll?
Would it be 1/6 X 1/18 X 1/18?
thanks.
Hi Debbe…Let’s break this down step by step:
### 1. **Rolling Doubles on the First Roll**
For a pair of dice:
– The probability of rolling any specific double (e.g., double 1s, double 2s, etc.) is \( \frac{1}{6} \).
– There are 6 possible doubles (1-1, 2-2, 3-3, 4-4, 5-5, 6-6).
– So, the probability of rolling any doubles is \( \frac{6}{36} = \frac{1}{6} \).
### 2. **Rolling the Same Doubles on the Next Roll**
Once you’ve rolled a specific double (say, double 4s), the probability of rolling the same doubles again (double 4s) is \( \frac{1}{36} \) because:
– There is only 1 outcome out of the 36 possible outcomes that matches the specific double you need.
### 3. **Rolling the Same Doubles on the Third Roll**
If you want to roll the same double again on the third roll, the probability is again \( \frac{1}{36} \).
### **Total Probability for All Three Rolls**
To find the probability of rolling the same doubles on three consecutive rolls, you multiply the probabilities of each event:
\[
\text{Probability} = \left(\frac{1}{6}\right) \times \left(\frac{1}{36}\right) \times \left(\frac{1}{36}\right) = \frac{1}{6} \times \frac{1}{36} \times \frac{1}{36} = \frac{1}{7776}
\]
### Summary
– The probability of rolling doubles on the first roll, followed by the **same** doubles on the next roll, is \( \frac{1}{6} \times \frac{1}{36} = \frac{1}{216} \).
– The probability of rolling the same doubles on the third roll as well is \( \frac{1}{216} \times \frac{1}{36} = \frac{1}{7776} \).
So the calculation you were thinking of is close, but the correct probability for the three identical doubles in a row is \( \frac{1}{7776} \).
So that means that rolling doubles is slightly less than rolling a total of 7 (16.67%), which is the most common outcome. So you’ll be relatively rolling a lot of doubles on your 2d6.
Very interesting…
Hi Brian…Yes, rolling doubles on two six-sided dice (2d6) is indeed an interesting probability topic! Here’s a quick breakdown:
### Probability of Rolling Doubles
Doubles occur when both dice show the same number: `(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)`. That’s 6 possible outcomes.
The total number of outcomes when rolling two dice is \( 6 \times 6 = 36 \).
So, the probability of rolling doubles is:
\[
\frac{6}{36} = 16.67\% \quad \text{(or 1/6)}.
\]
### Rolling a Total of 7
To roll a total of 7, you need any of these pairs:
– \( (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) \).
That’s 6 combinations as well. Since the total outcomes are the same (36), the probability of rolling a total of 7 is also:
\[
\frac{6}{36} = 16.67\% \quad \text{(or 1/6)}.
\]
### Why Does 7 Feel More Common?
While the probability of rolling a 7 and rolling doubles are the same, the *perception* often differs. This is because:
1. Rolling doubles feels less “common” because it requires a very specific match between the dice.
2. A total of 7 arises from different pairings of dice values, making it seem more diverse and frequent.
### Fun Fact
The distribution of totals when rolling 2d6 creates a bell-shaped curve, with 7 being the most common total. Doubles are evenly spaced probabilities, occurring at 1/6 each, so they don’t have the same “peak” as the total 7.