Question
Sherri has a drawer full of socks, all of which are identical except for the color. The drawer contains only white socks and blue socks.
Sherri determines that the probability of choosing a white pair of socks is 0.65. She states that since the drawer only has white socks and blue socks, the probability of randomly choosing a blue pair of socks must be the same as the probability of randomly choosing a white pair of socks.
Explain why Sherri's reasoning is not correct.
Determine the correct probability of randomly choosing a blue pair of socks.
Sherri determines that the probability of choosing a white pair of socks is 0.65. She states that since the drawer only has white socks and blue socks, the probability of randomly choosing a blue pair of socks must be the same as the probability of randomly choosing a white pair of socks.
Explain why Sherri's reasoning is not correct.
Determine the correct probability of randomly choosing a blue pair of socks.
Answers
GPT-4o mini
Sherri's reasoning is incorrect because the probabilities of choosing a white pair of socks and a blue pair of socks cannot be assumed to be equal simply because the only colors in the drawer are white and blue. Instead, the probabilities are contingent upon the proportions of each sock color in the drawer.
To clarify, let's denote the probabilities involved.
Let:
- \( p \) be the probability of choosing a white sock,
- \( 1 - p \) be the probability of choosing a blue sock.
Given that the probability of choosing a pair of white socks is \( 0.65 \), we can express that probability in terms of \( p \):
\[
P(\text{white pair}) = p^2 = 0.65.
\]
To find \( p \), we can solve for \( p \):
\[
p = \sqrt{0.65} \approx 0.8062.
\]
This means the probability of choosing a blue sock, which is \( 1 - p \), can now be calculated:
\[
1 - p \approx 1 - 0.8062 \approx 0.1938.
\]
Now we can calculate the probability of randomly choosing a blue pair of socks. The probability for a pair of blue socks is defined as:
\[
P(\text{blue pair}) = (1 - p)^2 \approx (0.1938)^2 \approx 0.0375.
\]
Thus, the correct probability of randomly choosing a blue pair of socks is approximately \( 0.0375 \) or \( 3.75\% \).
In summary:
- Sherri's assertion that the probability of choosing a blue pair of socks is the same as for a white pair is incorrect because they are dependent on the respective proportions of white and blue socks.
- The probability of randomly choosing a blue pair of socks is approximately \( 0.0375 \).
To clarify, let's denote the probabilities involved.
Let:
- \( p \) be the probability of choosing a white sock,
- \( 1 - p \) be the probability of choosing a blue sock.
Given that the probability of choosing a pair of white socks is \( 0.65 \), we can express that probability in terms of \( p \):
\[
P(\text{white pair}) = p^2 = 0.65.
\]
To find \( p \), we can solve for \( p \):
\[
p = \sqrt{0.65} \approx 0.8062.
\]
This means the probability of choosing a blue sock, which is \( 1 - p \), can now be calculated:
\[
1 - p \approx 1 - 0.8062 \approx 0.1938.
\]
Now we can calculate the probability of randomly choosing a blue pair of socks. The probability for a pair of blue socks is defined as:
\[
P(\text{blue pair}) = (1 - p)^2 \approx (0.1938)^2 \approx 0.0375.
\]
Thus, the correct probability of randomly choosing a blue pair of socks is approximately \( 0.0375 \) or \( 3.75\% \).
In summary:
- Sherri's assertion that the probability of choosing a blue pair of socks is the same as for a white pair is incorrect because they are dependent on the respective proportions of white and blue socks.
- The probability of randomly choosing a blue pair of socks is approximately \( 0.0375 \).