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

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

\(\frac{1}{14}\)

Step 1: Understand the Problem

We are given 50 tickets numbered from 00 to 49. We need to find the probability that the sum of digits of a selected ticket is 8, given that the product of the digits is zero. This means one of the digits must be 0.

Step 2: Condition of Product Being Zero

For the product of the two digits on the ticket to be zero, at least one of the digits must be 0. Considering numbers from 00 to 49, identify tickets where either the tens digit or the units digit is 0. This gives us the tickets: 00, 01, 02, 03, 04, 05, 06, 07, 08, 09, 10, 20, 30, 40.

Step 3: Condition of Sum Being Eight

From the list of tickets identified in Step 2, check which tickets have a sum of digits equal to 8. This occurs in the tickets: 08, 80, 17, 71, 26, 62, 35, 53, 44, but only 08 applies because one digit needs to be 0 (i.e., its product is zero). However, ticket 17, 26, 35, 44, practically cannot be formed. Thus, disregard non-zero products inconvenient cases.

Step 4: Counting Favorable Outcomes

The ticket among the qualified list (00, 01, 02, 03, 04, 05, 06, 07, 08, 09, 10, 20, 30, 40) where the sum of digits is 8 is 08. Hence, there is 1 favorable ticket.

Step 5: Counting Total Possible Outcomes

From Step 2, we found tickets 00, 01, 02, 03, 04, 05, 06, 07, 08, 09, 10, 20, 30, 40; hence there are 14 possible outcomes.

Step 6: Calculate the Probability

Probability is calculated as the number of favorable outcomes divided by the total number of possible outcomes. Thus, probability is \( \frac{1}{14} \).

Key Concepts

Conditional Probability

Conditional probability is the probability of an event occurring given that another event has already occurred. In general, if we have two events, A and B, the conditional probability of A given B is expressed as \( P(A|B) \). This is calculated using the formula:
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]
Where \(P(A \cap B)\) is the probability that both A and B occur, and \(P(B)\) is the probability that B occurs.
In the context of the exercise, the event A is "the sum of the digits is 8," whereas the event B is "the product of the digits is zero". Thus, we determine which tickets satisfy both conditions: product being zero, and sum being eight.
This step of differentiating between events is essential because it enables us to focus only on relevant cases, simplifying the probability calculation.
Working with conditional probability strengthens understanding of scenarios that involve dependencies among events.

Discrete Mathematics

Discrete mathematics involves mathematical structures that are fundamentally discrete, not continuous. This means the focus is on countable items, such as whole numbers, logical statements, and graphs.
Probabilistic computations often involve discrete structures like the finite set of numbered tickets from 00 to 49 in our exercise. Discrete concepts are vital in understanding how we map real-world conditions into numeric outcomes.
This mathematical approach is useful because it allows us to distinguish between separate possibilities. In our problem, we dealt with a finite set of outcomes and conditions (e.g., products and sums of digits), common in discrete mathematical problems.
Discrete mathematics helps us model and solve problems using finite items, which is practical and extensively applied in computer science, cryptography, and combinatorics.

Combinatorics

Combinatorics is a branch of mathematics dealing with the study of counting, arrangement, and combination of objects. In this exercise, combinatorics is applied to count the number of tickets that meet certain conditions.
The key combinatorial task here involves assessing how many tickets can meet both a specified sum of digits and product condition. First, we determined any ticket with a zero digit (product of digits is zero) and then checked which of these sum to eight.
This_selection process naturally involves combinatorial reasoning, where we determine the size of sets and subsets, such as those meeting particular criteria.
Often, basic principles of permutations and combinations are used, but in this case, listing and counting scenarios sufficed due to the small size. Combinatorics serves as a foundational tool for efficient counting and logical arrangement design.