Problem 32 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

One ticket is selected at random from 50 tickets numbered \(00,01,02, \ldots, 49 .\) The probability that the sum of the digits on the selected ticket is 8 , given that the product of these digits is zero, equals [AIEEE-2009] (a) \(\frac{1}{14}\) (b) \(\frac{1}{7}\) (c) \(\frac{5}{14}\) (d) \(\frac{1}{50}\)Solution (a) \(S=(00,01,02, \ldots, 49)\) Let \(A\) be the even that sum of the digits on the selected ticket is 8 then \(A=\\{08,17,26,35,44\\}\) Let \(B\) be the event that the product of the digits is zero. $$ \begin{aligned} &B=\\{00,01,02,03, \ldots, 09,10,20,30,40\\} \\ &A \cap B=\\{08\\} \\ &P(A / B)=\frac{P(A \cap B)}{P(B)}=\frac{n(A \cap B)}{n(B)}=\frac{1}{14} \end{aligned} $$

Step-by-Step Solution

Answer

The probability is \( \frac{1}{14} \).

Step 1: Understand the Problem

We're given 50 tickets numbered from 00 to 49. We need to find the probability that the sum of the digits on a selected ticket is 8, given that the product of these digits is zero.

Step 2: Define Relevant Events

Let \( A \) be the event where the sum of the digits on the selected ticket is 8, and \( B \) be the event where the product of the digits on the ticket is zero.

Step 3: List Possible Outcomes for Event A

Event \( A \) represents the condition where the sum of the digits equals 8. Possible tickets are: 08 (0+8), 17 (1+7), 26 (2+6), 35 (3+5), and 44 (4+4).

Step 4: List Possible Outcomes for Event B

Event \( B \) includes tickets where the product of the digits is zero. These tickets include any number with at least one digit as zero: 00 to 09, 10, 20, 30, and 40.

Step 5: Determine Outcomes in Both A and B

Find tickets where both events happen, i.e., \( A \cap B \). This is where the sum is 8 and the product is zero: only ticket 08 satisfies this (0+8).

Step 6: Calculate the Probability

Event \( B \) has 14 outcomes (00 to 09, 10, 20, 30, 40). Thus, \( n(B) = 14 \). Event \( A \cap B \) has 1 outcome, so \( n(A \cap B) = 1 \). The conditional probability \( P(A|B) = \frac{n(A \cap B)}{n(B)} = \frac{1}{14} \).

Key Concepts

Probability Theory

Probability theory is foundational in understanding whether certain outcomes are likely or unlikely. In the exercise involving the tickets, we're dealing with conditional probability. This is the probability of an event occurring, given that another event has already happened.

For the problem at hand, we define two events:

• Event A: The sum of the digits on the ticket is 8.
• Event B: The product of the digits is zero.

The challenge is to find the probability of Event A given Event B. In mathematical terms, this is written as \(P(A|B)\). The formula we use is:

• \(P(A|B) = \frac{P(A \cap B)}{P(B)}\)

This illustrates how probability changes under new conditions—in this case, that one of the digits is zero. Understanding this concept is critical in Probability Theory as it helps to model real-world situations where conditions affect outcomes.

Algebraic Problem Solving

Algebraic problem solving is a skill used to decode complex problems by reasoning and calculating step by step. In this problem, algebra helps us to list and identify possible outcomes.

Starting with the total set of outcomes, we use algebraic thinking to define subsets. Each subset matches one of the defined conditions. For instance:

• Event A (sum of digits = 8): We logically list all possible tickets: 08, 17, 26, 35, 44.
• Event B (product of digits = 0): This includes all tickets with at least one zero: 00, 01, 02, and so on up to 40.

Subsequently, we find the intersection where both conditions are true. In this problem, that means finding tickets that both sum to 8 and include a zero in one digit. The algebraic process triangulates events until we're left with a single valid ticket, which is 08.

Mathematics for Engineering Exams

Mathematics, particularly in engineering exams, often challenges students to demonstrate their problem-solving abilities using rigorous methods. This ticket problem is a classic example, testing one's grasp on probability and combinatorial logic.

To excel, students need to:

• Identify the conditions provided in the problem.
• Apply logical reasoning to determine which conditions overlap.
• Calculate outcomes methodically to isolate the correct solution.

Critical thinking and precision, as encouraged through solving this problem, are key components for succeeding in engineering mathematics. These skills ensure professionals can model and solve real-world challenges, balancing between theoretical understanding and practical application.