If \( E_1 \) and \( E_2 \) are two events associated with a random experiment such that: \( P(E_2) = 0.35 \), \( P(E_1 \cup E_2) = 0.85 \), \( P(E_1 \cap E_2) = 0.15 \), then \( P(E_1) \) is:
Concept
The probability of the union of two events is given by:
\[
P(A \cup B) = P(A) + P(B) - P(A \cap B)
\]
Given:
\[
P(E_2) = 0.35, \quad P(E_1 \cup E_2) = 0.85, \quad P(E_1 \cap E_2) = 0.15
\]
Substitute into the formula:
\[
P(E_1 \cup E_2) = P(E_1) + P(E_2) - P(E_1 \cap E_2)
\]
\[
0.85 = P(E_1) + 0.35 - 0.15
\]
\[
0.85 = P(E_1) + 0.20
\]
\[
P(E_1) = 0.85 - 0.20 = 0.65
\]
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