Geometry

Question Analysis
This question involves conditional probability in the context of a two - step game.
The main focus is on understanding the relationship between the first event (rolling an even number on a die) and the second event (drawing a heart from a deck of cards), and applying the conditional probability formula to find the probability of the second event given the first event has occurred.

Key Concept Explanation
Conditional probability measures the likelihood of an event happening when it is known that another event has already taken place.
The formula states that the probability of given is the ratio of the probability of both and occurring to the probability of occurring. In this problem, we need to calculate the probabilities of the individual events and their intersection to find the conditional probability.

Step-by-Step Solution
Calculate the probability of event (rolling an even number on a die):
A fair six - sided die has 6 possible outcomes: .
The even numbers are , so .
Calculate the probability of (rolling an even number on a die and drawing a heart from a deck of cards):
The probability of rolling an even number is , and the probability of drawing a heart from a standard deck of 52 cards is .
Since these two events are independent given that the die roll determines whether the card draw occurs,