Problem 164 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

One ticket is selected at random from 50 tickets numbered \(00,01,02, \ldots .49\). Then the probability that the sum of the digits on the selected ticket is 8 , given that the product of these digits is zero, equals [2009] (A) \(\frac{1}{14}\) (B) \(\frac{1}{7}\) (C) \(\frac{5}{14}\) (D) \(\frac{1}{50}\)

Step-by-Step Solution

Answer

The probability is \( \frac{1}{9} \).

Step 1: Understand the conditions

To solve the problem, we need to focus on two conditions: the sum of the digits is 8, and the product of the digits is zero. The condition of the product being zero implies that at least one of the digits is zero.

Step 2: Determine possible tickets with zero digit

The tickets are two-digit numbers ranging from 00 to 49. To satisfy the product being zero, either the tens digit or the ones digit must be zero. Thus, possible tickets include those with numbers like 00, 10, 20, 30, 40, 01, 02, ..., up to 09.

Step 3: Calculate the sum of digits equals 8

Given that one digit is zero, the other digit must be 8 to have a sum of 8. Therefore, the possible tickets satisfying both conditions are 08 and 80. However, since the maximum number is 49, only 08 satisfies all conditions.

Step 4: Count suitable tickets

From the list of tickets where one digit is zero (00 to 09 and multiples of 10 up to 40), we find only 08 meets both the conditions. There are 9 such numbers where one digit is zero (00, 01, 02, ..., 09).

Step 5: Calculate the probability

There is only 1 number out of the 9 that meets both conditions. Therefore, the probability is given by the formula \( \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{1}{9} \).

Key Concepts

Conditional Probability

Conditional probability is a crucial concept in probability theory that describes the likelihood of an event occurring, given that another event has already occurred. In the context of the exercise, we are asked to find the probability that the sum of the digits on a selected ticket is 8, assuming the product of these digits is zero.

To break it down, conditional probability can be mathematically expressed as:\[P(A|B) = \frac{P(A \cap B)}{P(B)}\]Here, \(P(A|B)\) is the probability of event \(A\) occurring given that \(B\) has occurred, \(P(A \cap B)\) is the probability that both events \(A\) and \(B\) occur, and \(P(B)\) is the probability of event \(B\) occurring.

In our problem:- Event \(A\) is that the sum of the digits is 8.- Event \(B\) is that the product of the digits is zero.

By focusing on the number of favorable tickets (like 08, where the sum is 8, and one digit is zero), and considering only those tickets where one digit is 0, we apply conditional probability to solve the exercise.

Combinatorics

Combinatorics deals with counting, arranging, and combining objects according to specific rules. In this exercise, combinatorics helps us identify tickets whose digit product yields zero and whose digit sum yields eight.

The product of digits being zero limits our choices, as at least one digit must be zero. Thus, we consider two-digit tickets, such as numbers ranging from 00 to 49, and focus on those where the tens or ones digit is zero.

With these constraints, we find two potential groups of tickets:

• Numbers where the tens digit is zero: 00, 01, 02, ..., 09.
• Numbers where the ones digit is zero: 10, 20, 30, 40.

To obtain a sum of 8, the other digit in the pair must be 8. However, the constraint that our numbers range to 49 limits our choice to 08 only. Combinatorics helps eliminate the impossible combinations to swiftly identify valid tickets.

Mathematical Reasoning

Mathematical reasoning involves logical thinking processes to solve problems, which is essential in tackling probability and combinatorics challenges. This exercise showcases the importance of applying logical reasoning to ensure all potential conditions and constraints are considered.

We begin by logically understanding that a zero product means one digit must be zero. Then, we need to deduce which combinations meet the additional condition where their sum is eight.

Logical steps taken include: - Listing all possible scenarios that fit the zero-product condition. - Narrowing down to combinations that could sum to eight. - Recognizing that numbers above 49 aren't valid, which constrains the solution further.

This logical approach ensures we do not overlook any valid possibilities and can effectively calculate the probability, considering only pertinent outcomes. Mathematical reasoning is crucial in confirming the final favorable outcome count is accurate, backed by logical deductions rather than mere assumptions.