# Why Does My Dice Probability Calculation Differ from Expected Results?

• B
• Thecla
In summary, the conversation discusses the calculation of the probability of throwing only one six when throwing a pair of dice. The formula for non-mutually exclusive events is used, but there is a mistake in counting the possibilities. The correct answer is 10/36, which can also be obtained by using the complement statement. The mistake was counting "at least one 6" instead of "only one 6".
Thecla
TL;DR Summary
What is probability of throwing one six in a throw of pair of dice?
I was trying to calculate the probability of throwing only one six when throwing a pair of dice.
Using the formula for non-mutually exclusive events : P(A)+P(B)-P(A,B) I get 1/6+1/6-1/36=11/36
but when I count all the 36 possibilities on paper I get 10/36 ways of getting only one 6. What am I doing wrong?

Seems like you are counting wrong.

You counted ##(6,6)## twice in ##P(A)+P(B)## but subtracted it only once.

FactChecker
Thanks for your help. So it is 10/36=1/6+1/6-2/36

Thecla said:
Thanks for your help. So it is 10/36=1/6+1/6-2/36
Yes. A common trick for those questions is to use the complement statement. We have ##5^2## cases without a ##6## plus ##(6,6)##, which leaves ##10## positive cases.
\begin{align*}
P((\lnot A \wedge \lnot B)\vee (A\wedge B) )&=P(\lnot A \wedge \lnot B)+P(A\wedge B)\\&=P(\lnot A)\cdot P(\lnot B)+P(A)\cdot P(B)\\&=\dfrac{5}{6}\cdot\dfrac{5}{6}+\dfrac{1}{6}\cdot\dfrac{1}{6}\\&=\dfrac{26}{36}
\end{align*}

Simplest of all is $$p = \frac 1 6 \cdot \frac 5 6 + \frac 5 6 \cdot \frac 1 6 = \frac {10}{36}$$based on whether a six is thrown on the first die or not.

Dale said:
Seems like you are counting wrong.
Oops, sorry, I was wrong. I was counting “at least one 6” instead of “only one 6”

## 1. What is the probability of rolling a specific number on a single die?

The probability of rolling a specific number on a single die is 1/6 or approximately 16.67%. This is because there are six possible outcomes (numbers 1-6) and each outcome has an equal chance of occurring.

## 2. What is the probability of rolling a specific combination of numbers on two dice?

The probability of rolling a specific combination of numbers on two dice is 1/36 or approximately 2.78%. This is because there are 36 possible outcomes (6 x 6) and each outcome has an equal chance of occurring.

## 3. How does the probability change if you roll more than two dice?

The probability of rolling a specific combination of numbers increases as you roll more dice. For example, the probability of rolling a specific combination of numbers on three dice is 1/216 or approximately 0.46%. This is because there are 216 possible outcomes (6 x 6 x 6) and each outcome has an equal chance of occurring.

## 4. What is the probability of rolling a certain sum on two dice?

The probability of rolling a certain sum on two dice can be calculated by finding the number of ways the sum can occur and dividing it by the total number of possible outcomes. For example, the probability of rolling a sum of 7 is 6/36 or approximately 16.67% (6 possible ways to roll a 7 out of 36 total possible outcomes).

## 5. How does the probability change if you add more dice or change the number of sides on the dice?

The probability will change depending on the number of dice and the number of sides on each dice. The more dice you add, the higher the probability of rolling a certain sum or combination. Similarly, if you change the number of sides on the dice, the probability will also change. For example, if you use a 12-sided die instead of a 6-sided die, the probability of rolling a specific number or combination will be different.

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