Question
Max and Ava are using a spinner with the numbers 1-8 to determine how many push-ups to do in their workout. What is the probability that they will have an odd number of push-ups three times in a row?
Answers
GPT-4o mini
First, we need to determine the total number of outcomes possible when spinning the spinner. The spinner has the numbers 1 through 8, which gives us a total of 8 outcomes.
Next, we identify how many of these outcomes are odd: the odd numbers in this range are 1, 3, 5, and 7. Therefore, there are 4 odd outcomes.
The probability of spinning an odd number (an odd outcome) on a single spin can be calculated as follows:
\[
\text{Probability of odd number} = \frac{\text{Number of odd outcomes}}{\text{Total outcomes}} = \frac{4}{8} = \frac{1}{2}.
\]
Max and Ava need to spin the spinner three times, and we want to find the probability that all three spins result in an odd number. Since the spins are independent, we can multiply the probabilities for each individual spin together.
The probability of getting an odd number on three consecutive spins is:
\[
P(\text{odd three times}) = P(\text{odd}) \times P(\text{odd}) \times P(\text{odd}) = \left(\frac{1}{2}\right) \times \left(\frac{1}{2}\right) \times \left(\frac{1}{2}\right).
\]
Calculating this gives:
\[
P(\text{odd three times}) = \left(\frac{1}{2}\right)^3 = \frac{1}{8}.
\]
Thus, the probability that they will have an odd number of push-ups three times in a row is \(\boxed{\frac{1}{8}}\).
Next, we identify how many of these outcomes are odd: the odd numbers in this range are 1, 3, 5, and 7. Therefore, there are 4 odd outcomes.
The probability of spinning an odd number (an odd outcome) on a single spin can be calculated as follows:
\[
\text{Probability of odd number} = \frac{\text{Number of odd outcomes}}{\text{Total outcomes}} = \frac{4}{8} = \frac{1}{2}.
\]
Max and Ava need to spin the spinner three times, and we want to find the probability that all three spins result in an odd number. Since the spins are independent, we can multiply the probabilities for each individual spin together.
The probability of getting an odd number on three consecutive spins is:
\[
P(\text{odd three times}) = P(\text{odd}) \times P(\text{odd}) \times P(\text{odd}) = \left(\frac{1}{2}\right) \times \left(\frac{1}{2}\right) \times \left(\frac{1}{2}\right).
\]
Calculating this gives:
\[
P(\text{odd three times}) = \left(\frac{1}{2}\right)^3 = \frac{1}{8}.
\]
Thus, the probability that they will have an odd number of push-ups three times in a row is \(\boxed{\frac{1}{8}}\).