Question
You are rolling a number cube to help you pick the number of minutes you will run in each of your workout sets. Determine the probability of rolling an odd number three times in a row.
Answers
GPT-4o mini
A standard number cube (or die) has six faces, numbered from 1 to 6. The odd numbers on the die are 1, 3, and 5. Therefore, there are 3 odd numbers out of a total of 6 numbers.
To find the probability of rolling an odd number, we can use the formula:
\[
P(\text{odd number}) = \frac{\text{number of odd outcomes}}{\text{total outcomes}} = \frac{3}{6} = \frac{1}{2}
\]
Now, we want to determine the probability of rolling an odd number three times in a row. Since the rolls are independent events, we can multiply the probabilities together:
\[
P(\text{3 odd numbers}) = P(\text{odd}) \times P(\text{odd}) \times P(\text{odd}) = \left(\frac{1}{2}\right) \times \left(\frac{1}{2}\right) \times \left(\frac{1}{2}\right) = \left(\frac{1}{2}\right)^3 = \frac{1}{8}
\]
Thus, the probability of rolling an odd number three times in a row is
\[
\frac{1}{8}.
\]
To find the probability of rolling an odd number, we can use the formula:
\[
P(\text{odd number}) = \frac{\text{number of odd outcomes}}{\text{total outcomes}} = \frac{3}{6} = \frac{1}{2}
\]
Now, we want to determine the probability of rolling an odd number three times in a row. Since the rolls are independent events, we can multiply the probabilities together:
\[
P(\text{3 odd numbers}) = P(\text{odd}) \times P(\text{odd}) \times P(\text{odd}) = \left(\frac{1}{2}\right) \times \left(\frac{1}{2}\right) \times \left(\frac{1}{2}\right) = \left(\frac{1}{2}\right)^3 = \frac{1}{8}
\]
Thus, the probability of rolling an odd number three times in a row is
\[
\frac{1}{8}.
\]