Question

select the correct choice that completes the sentence below.

the notation for the probability of \\(e_2\\), given \\(e_1\\), is

• \\(p(e_2 \cap e_1)\\)
• \\(p(e_2 \text{ and } e_1)\\)
• \\(p(e_2 | e_1)\\)
• \\(p(e_2) | e_1\\)
• \\(p(e_1 | e_2)\\)

Explanation:

Identify the conditional probability statement

The statement asks for the notation representing "the probability of \(E_2\), given \(E_1\)".

Apply conditional probability notation

Conditional probability of an event \(B\) given an event \(A\) is standardly denoted as \(P(B|A)\).
Here, the target event is \(E_2\) and the given condition is \(E_1\).
Therefore, the notation is \(P(E_2|E_1)\).

Match with the given options

Comparing \(P(E_2|E_1)\) with the dropdown choices:

• \(P(E_2 \cap E_1)\) represents the intersection.
• \(P(E_2 \text{ and } E_1)\) represents the intersection.
• \(P(E_2|E_1)\) represents the conditional probability of \(E_2\) given \(E_1\).
• \(P(E_2)|E_1\) is incorrectly formatted.
• \(P(E_1|E_2)\) represents the conditional probability of \(E_1\) given \(E_2\).

The correct option is \(P(E_2|E_1)\).

Answer:

• (A) \(P(E_2 \cap E_1)\)
• (B) \(P(E_2 \text{ and } E_1)\)
• (C) \(P(E_2|E_1)\) (Correct answer)
• (D) \(P(E_2)|E_1\)
• (E) \(P(E_1|E_2)\)