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 probability of choosing a blue pair of socks does not have to be the same as the probability of choosing a white pair of socks. This is because the probabilities of selecting white pairs and blue pairs are based on the proportions of white and blue socks in the drawer. If there are different quantities of each color sock, their probabilities will differ.
Let's denote:
- \( p_W \): probability of choosing a white sock
- \( p_B \): probability of choosing a blue sock
Since there are only white and blue socks in the drawer, we have:
\[
p_W + p_B = 1
\]
The probability of picking a pair of white socks, given that picking any sock, is \( p_W^2 \), and the probability of picking a pair of blue socks is \( p_B^2 \).
Sherri states the probability of selecting a white pair is 0.65.
Thus, we have:
\[
p_W^2 = 0.65
\]
To find \( p_W \):
\[
p_W = \sqrt{0.65} \approx 0.8062
\]
Now substituting into the equation \( p_W + p_B = 1 \) to find \( p_B \):
\[
p_B = 1 - p_W \approx 1 - 0.8062 \approx 0.1938
\]
Now we can calculate the probability of choosing a blue pair of socks:
\[
p_B^2 \approx (0.1938)^2 = 0.0376
\]
So the correct probability of randomly choosing a blue pair of socks is approximately **0.0376**, or **3.76%**. Thus, it is different from the probability of choosing a white pair of socks, which is 0.65 (or 65%).
Let's denote:
- \( p_W \): probability of choosing a white sock
- \( p_B \): probability of choosing a blue sock
Since there are only white and blue socks in the drawer, we have:
\[
p_W + p_B = 1
\]
The probability of picking a pair of white socks, given that picking any sock, is \( p_W^2 \), and the probability of picking a pair of blue socks is \( p_B^2 \).
Sherri states the probability of selecting a white pair is 0.65.
Thus, we have:
\[
p_W^2 = 0.65
\]
To find \( p_W \):
\[
p_W = \sqrt{0.65} \approx 0.8062
\]
Now substituting into the equation \( p_W + p_B = 1 \) to find \( p_B \):
\[
p_B = 1 - p_W \approx 1 - 0.8062 \approx 0.1938
\]
Now we can calculate the probability of choosing a blue pair of socks:
\[
p_B^2 \approx (0.1938)^2 = 0.0376
\]
So the correct probability of randomly choosing a blue pair of socks is approximately **0.0376**, or **3.76%**. Thus, it is different from the probability of choosing a white pair of socks, which is 0.65 (or 65%).