Problem 20 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

One of the two events must occur. If the chance of one is \(2 / 3\) of the other, then odds in favour of the other are (a) \(2: 3\) (b) \(1: 3\) (c) \(3: 1\) (d) \(3: 2\)

Step-by-Step Solution

Answer

The odds in favour of the other event are (d) \(3:2\).

Step 1: Define the Events

Let's assume two events, Event A and Event B, with their respective probabilities of occurring being \( P(A) \) and \( P(B) \). Since one of the two events must occur, that implies \( P(A) + P(B) = 1 \), because the total probability of all outcomes is always 1.

Step 2: Establish the Relationship Between Probabilities

We are given that the chance of one event is \( \frac{2}{3} \) of the other. Assume \( P(A) = \frac{2}{3}P(B) \). This gives us a relationship between the probabilities of A and B.

Step 3: Express All Probabilities in Terms of One Variable

From \( P(A) = \frac{2}{3}P(B) \), substitute \( P(A) = \frac{2}{3}P(B) \) into the equation \( P(A) + P(B) = 1 \):\[ \frac{2}{3}P(B) + P(B) = 1 \]Combine like terms:\[ \frac{5}{3}P(B) = 1 \]

Step 4: Solve for Probability of Event B

Solve the equation \( \frac{5}{3}P(B) = 1 \):\[ P(B) = \frac{3}{5} \]

Step 5: Find the Probability of Event A

Now that we have the probability of Event B, find the probability of Event A:\[ P(A) = 1 - P(B) = 1 - \frac{3}{5} = \frac{2}{5} \]

Step 6: Calculate Odds in Favour of Event B

Odds in favor of Event B are given by \( \frac{P(B)}{P(A)} \), which measures the ratio of the probability of B to the probability of A:\[ \text{Odds in favor of B} = \frac{P(B)}{P(A)} = \frac{\frac{3}{5}}{\frac{2}{5}} = \frac{3}{2} \]

Step 7: Interpret Odds in Simplest Form

The odds \( \frac{3}{2} \) can be expressed in terms of a ratio, which is \(3:2\).

Key Concepts

Understanding Odds

Odds are a way to express the likelihood that a certain event will occur as compared to its non-occurrence. They are different from probability, though closely related. Odds are typically expressed in the form of a ratio, such as "3:2," which reads as "three to two." This means that, for every three occurrences of the event in question, there are two non-occurrences.

The concept of odds is widely used in fields like gambling, sports betting, and statistics. Here's how to understand odds in a clearer context:

• If the odds are in favor (e.g., 3:2), it means the event is more likely to happen than not.
• If the odds are against (e.g., 2:3), the event is less likely to happen.
• Odds can be transformed into probabilities using the formula: \( P(E) = \frac{a}{a+b} \), where \(a:b\) are the odds in favor.

To convert odds back into probability, use the total number of parts in odds for both the event happening and not happening. Understanding this ratio gives you a deeper appreciation of probabilities and can help in making better predictions.

Exploring Probability of Events

Probability is the measure of the likelihood that an event will occur. It is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. Probability values range from 0 to 1, where "0" indicates impossibility and "1" indicates certainty.

Whenever you are assessing the probability of an event, encounter some basic principles:

• Probability of a certain event: \( P(E) = 1 \).
• Probability of an impossible event: \( P(E) = 0 \).
• The sum of probabilities for all possible outcomes must equal 1.

This concept is foundational in all calculations involving likelihoods, as you always start by determining possible outcomes and calculate accordingly. For example, in the given exercise, we used the concept that the sum of probabilities of two mutually exclusive events must equal 1.

Understanding Conditional Probability

Conditional probability refers to the likelihood of an event occurring given that another event has already occurred. It is a fundamental concept in probability theory that helps in understanding relationships between events.

Consider this general formula for conditional probability:\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]This formula reads out as "the probability of event A occurring given that event B has occurred".

• The symbol \(P(A|B)\) is the conditional probability.
• \(P(A \cap B)\) represents the probability that both events A and B occur.
• \(P(B)\) is the prior probability of event B.

The idea is relevant in fields such as data science, economics, and medical diagnoses, where one has to make predictions based on certain given conditions or prior information. Deep comprehension of conditional probability enhances decision-making in scenarios where events are interdependent.