Problem 72 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(\bar{E}\) and \(\bar{F}\) are the complementary events of events \(E\) and \(F\) respectively and if \(0

Step-by-Step Solution

Answer

Options (A) and (D) are true.

Step 1: Understanding Complementary Events

Complementary events are events that cover all possible outcomes of an experiment. For instance, if \(E\) is an event, then \(\bar{E}\), the complement of \(E\), is an event that occurs when \(E\) does not occur. Thus, the probabilities associated with complementary events are \(P(E) + P(\bar{E}) = 1\). This holds true for any event \(F\) as well.

Step 2: Analyze Each Option

We need to see which of the event combinations involving \(E\), \(\bar{E}\), \(F\), \(\bar{F}\) is always equal to 1. Recall: \(P(A \mid B)\) is the probability of \(A\) given \(B\).

Step 3: Examine Option A

For option (A), we have \(P(E \mid F) + P(\bar{E} \mid F) = 1\). Given that \(\bar{E}\) is the complementary of \(E\), within set \(F\), these are complementary probabilities. Thus, since all outcomes are covered, the sum is indeed 1.

Step 4: Evaluate Option B

For option (B), \(P(E \mid F) + P(E \mid \bar{F})\) does not necessarily add up to 1. \(P(E \mid F)\) and \(P(E \mid \bar{F})\) refer to different events (conditional probabilities of \(E\) given \(F\) and \(\bar{F}\), respectively), so they don't complement each other within the same sample space or event.

Step 5: Evaluate Option C

For option (C), we have \(P(\bar{E} \mid F) + P(E \mid \bar{F})\). These represent different events too: \(E\) given \(\bar{F}\) and \(\bar{E}\) given \(F\), they do not relate as complements within the same conditional context so they sum to a value that is not constrained to be 1.

Step 6: Evaluate Option D

In option (D), \(P(E \mid \bar{F}) + P(\bar{E} \mid \bar{F})\) equates to 1 because within the conditional context of \(\bar{F}\), \(E\) and \(\bar{E}\) are complementary events and their probabilities must sum to 1.

Step 7: Identify the Correct Answer

The possibilities out of the four options that are structured to show the completeness of complementary probabilities sum to 1 are (A) and (D). These options respect the complementary principle within the context of their respective conditioning sets.

Key Concepts

Conditional Probability

Conditional probability is a fundamental concept in probability theory. It represents the likelihood of an event occurring given that another event has already occurred. This is typically described using the notation \(P(A \mid B)\), which reads as "the probability of \(A\) given \(B\)."
Understanding conditional probability is crucial because it allows for a more nuanced view of probability, one that considers the influence of certain conditions or pre-existing events on the probability of another event. Conditional probability is calculated using the formula:

• \(P(A \mid B) = \frac{P(A \cap B)}{P(B)}\) if \(P(B) > 0\)

This formula implies that we divide the probability of both events \(A\) and \(B\) occurring by the probability of event \(B\) itself.
In many real-world situations, calculating the probability of an event influenced by another event helps in decision-making, risk assessment, and predictive modeling. It narrows down possible outcomes to a subset where the condition holds true, providing a clearer picture of what might happen.

Complementary Events

Complementary events are pairs of outcomes that make up the entire sample space of an event. When one of these events occurs, the other does not, and vice versa. Mathematically, if \(E\) is an event, then \(\bar{E}\) is its complement.
An important property of complementary events is that their probabilities add up to 1:

• \(P(E) + P(\bar{E}) = 1\)

This property means that if you know the probability of one event, you automatically know the probability of its complement. For instance, in the roll of a die, the event of rolling an even number \(E\) and the event of rolling an odd number \(\bar{E}\) are complementary.
Understanding complementary events is essential because they allow us to assess the likelihood of an event by simply knowing what is not the event. This relationship can also help simplify complex probability problems by using the complement rule, which states:

• \(P(\bar{E}) = 1 - P(E)\)

This is particularly useful for probability calculations where determining \(P(E)\) directly is challenging, yet \(P(\bar{E})\) can be easily found.

Event Probabilities

Event probabilities are the backbone of probability theory. Each event in an experiment or sample space is assigned a probability, a number between 0 and 1, reflecting its likelihood of occurring.
Probabilities can be determined using different methods:

• Theoretical probability, which is based on the expectation due to symmetry or logical basis. For example, the probability of rolling a 3 on a fair die is \(\frac{1}{6}\).
• Empirical probability, which is based on experiment or historical data, calculated as \(\frac{\text{Number of successful outcomes}}{\text{Total number of trials}}\).
• Subjective probability, which derives from personal judgment or estimation.

Every event's probability must adhere to key axioms:

• Non-negativity: \(P(E) \geq 0\)
• Normalization: \(P(S) = 1\) for the entire sample space \(S\)
• Additivity: For mutually exclusive events \(A\) and \(B\), \(P(A \cup B) = P(A) + P(B)\)

These principles form the foundation upon which more complex probability theories are built. By understanding the probability of simple and compound events, one can navigate through more complicated probability scenarios with confidence.