The Deuce
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    heh. I asked chatgpt as well and got a different answer Archived Message

    Posted by snipes824 on February 22, 2025, 14:18:28, in reply to "According to ChatGPT...."

    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.

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