Math / Data & Statistics

Question\begin{tabular}{|l|l|} \hline What are the chances of them adding up to 6 or 8 ? & 0.167 \\ \hline What are the chances of them adding up to a prime number? & \\ \hline What are the chances of rolling a double (same number on each die)? \\ \hline What are the chances of rolling snake-eyes twice in a row? & \\ \hline \end{tabular}

Studdy Solution

STEP 1

What is this asking? We're rolling two dice, and we want to figure out the probabilities of getting certain sums or outcomes! Watch out! Don't forget that each die has 6 sides, numbered 1 to 6, and that each roll is independent!

STEP 2

1. Prime Sum Probability
2. Double Probability
3. Snake Eyes Probability

STEP 3

Alright, let's investigate the prime sums!
First, let's list all the possible sums we can get when rolling two dice.
The smallest sum is $1 + 1 = 2$ and the largest is $6 + 6 = 12$ , so our possible sums are 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12.

STEP 4

Now, let's identify the prime numbers in that list.
Remember, a prime number is a whole number greater than 1 that has only two divisors: 1 and itself.
Looking at our list, the prime sums are 2, 3, 5, 7, and 11.

STEP 5

Time to figure out how many ways we can roll each of these prime sums.
We can get a sum of 2 in only one way: $1 + 1$ .
A sum of 3 can be rolled in two ways: $1 + 2$ and $2 + 1$ .
A sum of 5 can be rolled in four ways: $1 + 4$ , $2 + 3$ , $3 + 2$ , and $4 + 1$ .
A sum of 7 can be rolled in six ways: $1 + 6$ , $2 + 5$ , $3 + 4$ , $4 + 3$ , $5 + 2$ , and $6 + 1$ .
Finally, an 11 can be rolled in two ways: $5 + 6$ and $6 + 5$ .

STEP 6

Let's add up the number of ways to get a prime sum: $1 + 2 + 4 + 6 + 2 = 15$ .
Since there are $6 \cdot 6 = 36$ total possible outcomes when rolling two dice, the probability of rolling a prime sum is $\frac{15}{36} = \frac{5}{12}$ .

STEP 7

Rolling doubles means both dice show the same number.
This can happen in six ways: $1 + 1$ , $2 + 2$ , $3 + 3$ , $4 + 4$ , $5 + 5$ , and $6 + 6$ .

STEP 8

With $6$ favorable outcomes and $36$ total possible outcomes, the probability of rolling doubles is $\frac{6}{36} = \frac{1}{6}$ .

STEP 9

Snake eyes means rolling two ones.
There's only one way to do this: $1 + 1$ .

STEP 10

The probability of rolling snake eyes once is $\frac{1}{36}$ .
Since each roll is independent, the probability of rolling snake eyes twice in a row is $\frac{1}{36} \cdot \frac{1}{36} = \frac{1}{1296}$ .

STEP 11

The probability of rolling a prime sum is $\frac{5}{12}$ .
The probability of rolling doubles is $\frac{1}{6}$ .
The probability of rolling snake eyes twice in a row is $\frac{1}{1296}$ .

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Studdy Solution

STEP 1

What is this asking? We're rolling two dice, and we want to figure out the probabilities of getting certain sums or outcomes! Watch out! Don't forget that each die has 6 sides, numbered 1 to 6, and that each roll is independent!

STEP 2

1. Prime Sum Probability
2. Double Probability
3. Snake Eyes Probability

STEP 3

Alright, let's investigate the prime sums!
First, let's list all the possible sums we can get when rolling two dice.
The smallest sum is $1 + 1 = 2$ and the largest is $6 + 6 = 12$ , so our possible sums are 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12.

STEP 4

Now, let's identify the prime numbers in that list.
Remember, a prime number is a whole number greater than 1 that has only two divisors: 1 and itself.
Looking at our list, the prime sums are 2, 3, 5, 7, and 11.

STEP 5

STEP 6

Let's add up the number of ways to get a prime sum: $1 + 2 + 4 + 6 + 2 = 15$ .
Since there are $6 \cdot 6 = 36$ total possible outcomes when rolling two dice, the probability of rolling a prime sum is $\frac{15}{36} = \frac{5}{12}$ .

STEP 7

Rolling doubles means both dice show the same number.
This can happen in six ways: $1 + 1$ , $2 + 2$ , $3 + 3$ , $4 + 4$ , $5 + 5$ , and $6 + 6$ .

STEP 8

With $6$ favorable outcomes and $36$ total possible outcomes, the probability of rolling doubles is $\frac{6}{36} = \frac{1}{6}$ .

STEP 9

Snake eyes means rolling two ones.
There's only one way to do this: $1 + 1$ .

STEP 10

The probability of rolling snake eyes once is $\frac{1}{36}$ .
Since each roll is independent, the probability of rolling snake eyes twice in a row is $\frac{1}{36} \cdot \frac{1}{36} = \frac{1}{1296}$ .

STEP 11

The probability of rolling a prime sum is $\frac{5}{12}$ .
The probability of rolling doubles is $\frac{1}{6}$ .
The probability of rolling snake eyes twice in a row is $\frac{1}{1296}$ .

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Studdy Solution

STEP 1

What is this asking? We're rolling two dice, and we want to figure out the probabilities of getting certain sums or outcomes! Watch out! Don't forget that each die has 6 sides, numbered 1 to 6, and that each roll is independent!

STEP 2

1. Prime Sum Probability2. Double Probability3. Snake Eyes Probability

STEP 3

Alright, let's **investigate** the prime sums!First, let's **list** all the possible sums we can get when rolling two dice.The smallest sum is 1+1=21 + 1 = 21+1=2 and the largest is 6+6=126 + 6 = 126+6=12, so our possible sums are 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12.

STEP 4

Now, let's **identify** the prime numbers in that list.Remember, a prime number is a whole number greater than 1 that has only two divisors: 1 and itself.Looking at our list, the prime sums are **2, 3, 5, 7, and 11**.

STEP 5

STEP 6

Let's **add up** the number of ways to get a prime sum: 1+2+4+6+2=151 + 2 + 4 + 6 + 2 = 151+2+4+6+2=15.Since there are 6⋅6=366 \cdot 6 = 366⋅6=36 total possible outcomes when rolling two dice, the probability of rolling a prime sum is 1536=512\frac{15}{36} = \frac{5}{12}3615​=125​.

STEP 7

Rolling doubles means both dice show the same number.This can happen in six ways: 1+11 + 11+1, 2+22 + 22+2, 3+33 + 33+3, 4+44 + 44+4, 5+55 + 55+5, and 6+66 + 66+6.

STEP 8

With 666 favorable outcomes and 363636 total possible outcomes, the probability of rolling doubles is 636=16\frac{6}{36} = \frac{1}{6}366​=61​.

STEP 9

Snake eyes means rolling two ones.There's only one way to do this: 1+11 + 11+1.

STEP 10

The probability of rolling snake eyes once is 136\frac{1}{36}361​.Since each roll is independent, the probability of rolling snake eyes *twice* in a row is 136⋅136=11296\frac{1}{36} \cdot \frac{1}{36} = \frac{1}{1296}361​⋅361​=12961​.

STEP 11

The probability of rolling a prime sum is 512\frac{5}{12}125​.The probability of rolling doubles is 16\frac{1}{6}61​.The probability of rolling snake eyes twice in a row is 11296\frac{1}{1296}12961​.

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1. Prime Sum Probability
2. Double Probability
3. Snake Eyes Probability

Alright, let's investigate the prime sums!
First, let's list all the possible sums we can get when rolling two dice.
The smallest sum is $1 + 1 = 2$ and the largest is $6 + 6 = 12$ , so our possible sums are 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12.

Now, let's identify the prime numbers in that list.
Remember, a prime number is a whole number greater than 1 that has only two divisors: 1 and itself.
Looking at our list, the prime sums are 2, 3, 5, 7, and 11.

Let's add up the number of ways to get a prime sum: $1 + 2 + 4 + 6 + 2 = 15$ .
Since there are $6 \cdot 6 = 36$ total possible outcomes when rolling two dice, the probability of rolling a prime sum is $\frac{15}{36} = \frac{5}{12}$ .

Rolling doubles means both dice show the same number.
This can happen in six ways: $1 + 1$ , $2 + 2$ , $3 + 3$ , $4 + 4$ , $5 + 5$ , and $6 + 6$ .

With $6$ favorable outcomes and $36$ total possible outcomes, the probability of rolling doubles is $\frac{6}{36} = \frac{1}{6}$ .

Snake eyes means rolling two ones.
There's only one way to do this: $1 + 1$ .

The probability of rolling snake eyes once is $\frac{1}{36}$ .
Since each roll is independent, the probability of rolling snake eyes twice in a row is $\frac{1}{36} \cdot \frac{1}{36} = \frac{1}{1296}$ .

The probability of rolling a prime sum is $\frac{5}{12}$ .
The probability of rolling doubles is $\frac{1}{6}$ .
The probability of rolling snake eyes twice in a row is $\frac{1}{1296}$ .