The Deuce: Interesting. Good job long snapper*

so I copied your entire answer and said a friend asked the same question and got a different answer, which one is correct? I also pressed the "reason" button this time, and it realized that both were actually wrong and longsnapper was right lol:

Reasoned about probability of three of a kind for 48 seconds
Both answers contain errors. In fact, when “three‐of‐a‐kind” is defined as having at least three dice showing the same number (with 3 dice this means all dice match, and with 4 dice it means either exactly three match or all four match), the correct calculations are as follows:

Rolling 3 Dice
There are a total of
6
3
=
216
6
3
=216 outcomes. To have three of a kind (all dice the same), there are exactly 6 outcomes (one for each number):

P
(
3
dice
)
=
6
216
=
1
36
≈
2.78
%
P(3 dice)=
216
6

=
36
1

≈2.78%
Note: Some earlier answers mistakenly treated the “free” first die as having a
1
/
6
1/6 chance instead of 1.

Rolling 4 Dice
There are
6
4
=
1296
6
4
=1296 outcomes. You can get “three of a kind” (i.e. at least three matching) in two ways:

Exactly three matching and one different:
Choose the number that appears three times: 6 ways.
Choose which 3 of the 4 dice show that number:
(
4
3
)
=
4
(
3
4

)=4 ways.
The remaining die must show a different number (5 choices).
Number of outcomes:

6
×
4
×
5
=
120
6×4×5=120
All four matching:
There are 6 outcomes (one for each number).
Total successful outcomes:

120
+
6
=
126
120+6=126
So, the probability is:

P
(
4
dice
)
=
126
1296
≈
9.72
%
P(4 dice)=
1296
126

≈9.72%
Comparing the Two
3 Dice: ~2.78%
4 Dice: ~9.72%
This means that rolling 4 dice increases your chance of getting three of a kind by an absolute 6.94 percentage points. In relative terms, you’re about
9.72
2.78
≈
3.5
2.78
9.72

≈3.5 times more likely (roughly a 250% increase relative to the 3‐dice chance).

Conclusion
Neither the friend’s answer (which yielded about 16.67% for 3 dice and 28.7% for 4 dice) nor the previous assistant’s answer (with 16.67% vs. 19.44%) is correct if we use standard probability with fair six‐sided dice. The correct probabilities are approximately 2.78% with 3 dice and 9.72% with 4 dice.

Thus, rolling 4 dice increases your chance of getting three of a kind by roughly 3.5 times compared to rolling 3 dice.

Previous Message

Let's calculate the probability of rolling **three of a kind** using **three dice** and **four dice** to determine how much the odds increase.

### **Case 1: Rolling 3 Dice**
We need exactly **three** dice to show the same number.

1. Pick a number (1-6) that will be the triplet → Probability = \( \frac{1}{6} \).
2. The first die can be any number → Probability = \(1\).
3. The second die must match the first → Probability = \( \frac{1}{6} \).
4. The third die must also match → Probability = \( \frac{1}{6} \).

Thus, the probability of rolling three of a kind exactly is:
\[
6 \times \left(\frac{1}{6} \times \frac{1}{6}\right) = \frac{6}{36} = \frac{1}{6} \approx 16.67\%
\]

### **Case 2: Rolling 4 Dice**
We need at least **three** of the four dice to show the same number.

#### **Scenario 1: Exactly 3 Matching, 1 Different**
1. Pick the matching number (1-6) → \( \frac{1}{6} \).
2. Choose which three dice will match (ways to choose 3 out of 4) → \( \binom{4}{3} = 4 \).
3. The three matching dice each roll that number → \( \left(\frac{1}{6}\right)^2 \).
4. The fourth die rolls a different number (5 choices) → \( \frac{5}{6} \).

\[
6 \times 4 \times \left(\frac{1}{6} \times \frac{1}{6}\right) \times \frac{5}{6} = \frac{120}{1296} \approx 9.26\%
\]

#### **Scenario 2: All 4 Matching**
1. Pick the number → \( \frac{1}{6} \).
2. All four dice must roll that number → \( \left(\frac{1}{6}\right)^3 \).

\[
6 \times \left(\frac{1}{6} \times \frac{1}{6} \times \frac{1}{6}\right) = \frac{6}{216} = \frac{1}{36} \approx 2.78\%
\]

### **Total Probability (Rolling 4 Dice)**
Adding both scenarios together:

\[
9.26\% + 2.78\% = 12.04\% \text{ (extra chance with 4th die)}
\]

\[
16.67\% + 12.04\% = 28.7\%
\]

### **Odds Increase**
\[
\frac{28.7\%}{16.67\%} \approx 1.72
\]

So, rolling **4 dice instead of 3 increases your odds by about 72%** of rolling three of a kind.