Gardner Dawson
06/22/2023 · Primary School
Given events \( C \) and \( D \) with probabilities \( P(C)=0.3, P(D)=0.2 \), and \( P(C \) and \( D)=0.1 \), are \( C \) and \( D \) independent?
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Step-by-step Solution
To determine whether events \( C \) and \( D \) are independent, we need to check if the following condition holds:
\[
P(C \cap D) = P(C) \cdot P(D)
\]
We are given the following probabilities:
- \( P(C) = 0.3 \)
- \( P(D) = 0.2 \)
- \( P(C \cap D) = 0.1 \)
Now, we can calculate \( P(C) \cdot P(D) \):
\[
P(C) \cdot P(D) = 0.3 \cdot 0.2 = 0.06
\]
Next, we compare \( P(C \cap D) \) with \( P(C) \cdot P(D) \):
- \( P(C \cap D) = 0.1 \)
- \( P(C) \cdot P(D) = 0.06 \)
Since \( P(C \cap D) = 0.1 \) is not equal to \( P(C) \cdot P(D) = 0.06 \), we conclude that:
\[
P(C \cap D) \neq P(C) \cdot P(D)
\]
Therefore, events \( C \) and \( D \) are **not independent**.
Quick Answer
Events \( C \) and \( D \) are not independent.
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