Problem 11 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

A biased die is tossed and the respective probabilities for various faces to turn up are given below: \(\begin{array}{lcccccc}\text { Face } & : 1 & 2 & 3 & 4 & 5 & 6 \\ \text { Probability: } & 0.1 & 0.24 & 0.19 & 0.18 & 0.15 & 0.14\end{array}\) If an even face has turned up, then the probability that it is face 2 or face 4 is (a) \(0.25\) (b) \(0.42\) (c) \(0.75\) (d) \(0.9\)

Step-by-Step Solution

Answer

(c) 0.75

Step 1: Identify Even Faces

First, identify the faces of the die that are even numbers. The even faces on a die are 2, 4, and 6.

Step 2: Calculate Probability of Even Faces

Add the probabilities of the even faces turning up. For face 2, the probability is 0.24; for face 4 it is 0.18; for face 6 it is 0.14. Sum these probabilities to find the total probability of an even face turning up: \( P( ext{Even face}) = 0.24 + 0.18 + 0.14 = 0.56 \).

Step 3: Calculate Combined Probability for Faces 2 and 4

Calculate the total probability of face 2 or face 4, which are the desired outcomes. This is done by adding their individual probabilities:\( P( ext{Face 2 or Face 4}) = 0.24 + 0.18 = 0.42 \).

Step 4: Apply Conditional Probability Formula

Since we want the probability of face 2 or 4 given that an even face has already turned up, apply the conditional probability formula:\( P( ext{Face 2 or 4 | Even face}) = \frac{P( ext{Face 2 or 4})}{P( ext{Even face})} \).Substitute the values calculated earlier:\( P( ext{Face 2 or 4 | Even face}) = \frac{0.42}{0.56} = 0.75 \).

Step 5: Select the Correct Option

The calculated conditional probability is 0.75. Hence, the correct option is (c) 0.75.

Key Concepts

Probability Theory Basics

Probability theory is a fascinating mathematical concept that deals with the likelihood of events happening. It's essential in a wide range of fields from finance to science. In simple terms, probability measures how likely something is to occur out of a set of possible outcomes. It is expressed as a number between 0 and 1, where 0 means the event will not occur and 1 means it will certainly happen.

When working with probability, we often deal with scenarios that involve rolling dice or drawing cards. Each of these setups involves random variables, or outcomes, each with a certain probability associated with them. The total probability of all possible outcomes must always equal 1. For instance, when rolling a die, you're dealing with six different outcomes, each representing a face of that die.

• Basic Formula: The probability of an event is calculated using the formula:
  \( P(E) = \frac{\text{Number of favourable outcomes}}{\text{Total number of possible outcomes}} \)

This framework allows us to calculate the likelihood of any outcome in a given set of possibilities.

Understanding a Biased Die

A biased die is a die that does not have an equal probability of landing on any given face. Unlike a fair die where each face has a probability of 1/6, a biased die's faces have different probabilities. This can be used to model real-world scenarios where outcomes have an inherent bias or skewed expectation.

• Unequal Probabilities: With a biased die, each number on the die has a specific probability of appearing. The sum of all face probabilities still equals 1. In our example, each face of the die has distinct probabilities assigned to it, such as 0.1 for face 1 and 0.24 for face 2. Unlike a fair die, you can't simply divide by 6 to find each face's probability.
• Importance in Problem Solving: Understanding how to work with biased dice is crucial in complex probability problems. It allows one to calculate conditional probabilities, which consider known conditions or events affecting outcomes, such as knowing an even face has turned up.

In practical terms, this kind of problem enhances understanding of more complex statistics beyond simple random events.

Role of Even Numbers in Probability

In probability problems, identifying even numbers is often key to simplifying the problem. This concept aligns well with problems involving dice, cards, or any set where even numbers have specific implications. Even numbers are integers divisible by 2, such as 2, 4, and 6.

When dealing with dice, even numbers often form part of conditions in probability questions. Determining the probability of rolling an even number or using even numbers as part of conditional calculations can lead to deeper insights into a problem.

• Even Numbers in Dice: On a standard six-sided die, the even numbers are 2, 4, and 6. Each has been pre-assigned a probability based on the context of biased or fair dice.
• Practical Calculation: To refine the probability calculations when an even number has appeared, one can calculate the sum of probabilities associated with even faces. Then, conditional probability is used to find the likelihood of further events, like getting a specific number once you know an even face came up. This step was crucial in our die problem, leading to the correct answer of 0.75.

Recognizing the role of even numbers helps to streamline complex probability scenarios.