Two fair dice are rolled simultaneously. Let X denote the...

Two fair dice are rolled simultaneously. Let $X$ denote the number on the first die and $Y$ denote the number on the second die. Then the value of $P(X+Y \leq 7 / Y \geq 5)$ is
(A) $\frac{1}{2}$
(B) $\frac{1}{3}$
(C) $\frac{1}{4}$
(D) $\frac{1}{6}$

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Key Concepts

Conditional Probability

This concept involves the probability of an event occurring given that another event has already occurred. It is calculated as the probability of the joint occurrence of both events divided by the probability of the given event. Understanding conditional probability is essential for updating probability assessments when new information is available.

Counting in Finite Sample Spaces

In problems involving discrete, equally likely outcomes, such as rolling dice, counting techniques are used to enumerate the total number of outcomes and the number of favorable outcomes for events. This systematic approach helps in calculating probabilities by forming ratios of favorable outcomes to the total outcomes.

Joint Probability of Independent Events

When dealing with multiple independent events, such as separate dice rolls, the joint probability of two events is the product of their individual probabilities. This principle simplifies the analysis of compound events and is useful when calculating probabilities in combined trials.

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