When a fair die is thrown twice, let (a, b) denote the...

When a fair die is thrown twice, let $(a, b)$ denote the outcome in which the first throw shows $a$ and the second shows $b$. Further, let $A, B$ and $C$ be the following events: $A=I(a, b) \mid a$ is odd $\}$. $B=\{(a, b) \mid b$ is odd $\}$ and $C=\{(a, b) \mid a+b$ is odd $\}$. Then (a) $P(A \cap B)=1 / 4$ (b) $P(B \cap C)=1 / 4$ (c) $P(A \cap C)=1 / 4$ (d) $P(A \cap B \cap C)=0$

When a fair die is thrown twice, let $(a, b)$ denote the outcome in which the first throw shows $a$ and the second shows $b$. Further, let $A, B$ and $C$ be the following events: $A=I(a, b) \mid a$ is odd $\}$. $B=\{(a, b) \mid b$ is odd $\}$ and $C=\{(a, b) \mid a+b$ is odd $\}$. Then
(a) $P(A \cap B)=1 / 4$
(b) $P(B \cap C)=1 / 4$
(c) $P(A \cap C)=1 / 4$
(d) $P(A \cap B \cap C)=0$

Key Concepts

Parity Concepts in Numbers

Parity refers to whether an integer is odd or even. In probability problems involving dice, understanding the parity of outcomes is important, especially when events are defined based on whether numbers are odd or even, or when sums of outcomes are considered. Knowledge of parity rules helps in determining which outcomes satisfy conditions like having an odd sum.

Intersection of Events

The intersection of events involves outcomes that satisfy all given conditions simultaneously. In probability, computing the intersection is about identifying and counting outcomes that are common to two or more events. This concept is critical when calculating the likelihood that multiple specific conditions occur at the same time, using techniques such as direct enumeration or by leveraging properties like independence.

Sample Space

The sample space in a probability problem refers to the set of all possible outcomes of the experiment. In the context of throwing dice, it includes every ordered pair of outcomes that can occur. A clear understanding of the sample space is essential for calculating probabilities, as the probability of any event is determined by the ratio of the number of favorable outcomes to the total number of outcomes.

Probability of Independent Events

When an experiment involves independent events, such as sequential dice throws, the outcome of one event does not affect the outcome of another. This concept allows the probabilities of individual events to be multiplied to find the probability of their joint occurrence. Recognizing independence is key to correctly computing the probability of multiple events happening together.

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