Question
You are rolling a number cube (1-6) to help you pick the number of minutes you will run in each of your workout sets. What is the probability of rolling a number less than 5 three times in a row?
(1 point)
Responses
8278 over 27
232 thirds
212 over 1
12161 over 216
(1 point)
Responses
8278 over 27
232 thirds
212 over 1
12161 over 216
Answers
GPT-4o mini
To find the probability of rolling a number less than 5 (which means rolling a 1, 2, 3, or 4) on a number cube (which has the numbers 1 through 6), we first calculate the probability of rolling a number less than 5.
There are 4 favorable outcomes (1, 2, 3, 4) out of 6 possible outcomes. Therefore, the probability \(P\) of rolling a number less than 5 is:
\[
P(\text{less than 5}) = \frac{4}{6} = \frac{2}{3}
\]
Now, we want the probability of this happening three times in a row. Since the rolls are independent, we can multiply the probabilities together:
\[
P(\text{less than 5 three times in a row}) = P(\text{less than 5}) \times P(\text{less than 5}) \times P(\text{less than 5}) = \left(\frac{2}{3}\right)^3
\]
Calculating this gives:
\[
\left(\frac{2}{3}\right)^3 = \frac{2^3}{3^3} = \frac{8}{27}
\]
Now, let's find the fraction from the given responses:
1. \( \frac{8278}{27} \): Not a simple form, unrelated
2. \( \frac{232}{3} \): Not applicable, unrelated.
3. \( \frac{212}{1} \): Unrelated.
4. \( \frac{12161}{216} \): Not a match.
However, from our calculations, the correct answer for the probability of rolling a number less than 5 three times in a row is:
\[
\frac{8}{27}
\]
None of the provided options directly match this. This might indicate a mistake in the options provided. Please double-check the options given. The calculated probability based on standard dice rolling conventions is \( \frac{8}{27} \).
There are 4 favorable outcomes (1, 2, 3, 4) out of 6 possible outcomes. Therefore, the probability \(P\) of rolling a number less than 5 is:
\[
P(\text{less than 5}) = \frac{4}{6} = \frac{2}{3}
\]
Now, we want the probability of this happening three times in a row. Since the rolls are independent, we can multiply the probabilities together:
\[
P(\text{less than 5 three times in a row}) = P(\text{less than 5}) \times P(\text{less than 5}) \times P(\text{less than 5}) = \left(\frac{2}{3}\right)^3
\]
Calculating this gives:
\[
\left(\frac{2}{3}\right)^3 = \frac{2^3}{3^3} = \frac{8}{27}
\]
Now, let's find the fraction from the given responses:
1. \( \frac{8278}{27} \): Not a simple form, unrelated
2. \( \frac{232}{3} \): Not applicable, unrelated.
3. \( \frac{212}{1} \): Unrelated.
4. \( \frac{12161}{216} \): Not a match.
However, from our calculations, the correct answer for the probability of rolling a number less than 5 three times in a row is:
\[
\frac{8}{27}
\]
None of the provided options directly match this. This might indicate a mistake in the options provided. Please double-check the options given. The calculated probability based on standard dice rolling conventions is \( \frac{8}{27} \).