Problem 41 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

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) \(\frac{1}{7}\) (b) \(\frac{5}{14}\) (c) \(\frac{1}{50}\) (d) \(\frac{1}{14}\)

Step-by-Step Solution

Answer

The probability is \(\frac{1}{7}\).

Step 1: Understanding the problem

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

Step 2: Identify the condition for the product

The condition provided is that the product of the digits is zero. For a product of the two digits (x, y) to be zero, at least one of the digits must be zero. This means the ticket number should include the digit '0'.

Step 3: Identify numbers with zero

The numbers between 00 and 49 that have at least one digit as zero are: 00, 01, 02, 03, 04, 05, 06, 07, 08, 09, 10, 20, 30, and 40. These comprise one digit set to zero and the other digit ranging from 0 to 9 or 1 to 4 in tens.

Step 4: Filter based on the sum of digits

Out of these 14 numbers, we need to find which have a sum of digits equal to 8. The numbers are 08, 08, and 80 (from permutations like 02-20, 05-50, and 32).

Step 5: Counting favorable and possible outcomes

There are 2 numbers that satisfy both conditions. Total outcomes possible under the condition of having one zero digit are 14 (from Step 3).

Step 6: Calculate the probability

The probability is the number of favorable outcomes over the number of possible outcomes. Therefore, the probability is given by \( \frac{2}{14} = \frac{1}{7} \).

Key Concepts

Conditional Probability

Conditional probability is a fundamental concept in probability theory.It helps us calculate the likelihood of an event occurring, given that another event has already happened.
In our exercise, we are asked to find the probability that the sum of the digits on a ticket is 8, given that the product of these digits is zero.This is symbolically represented as \( P(A \mid B) \), where \( A \) is the event "sum of digits is 8" and \( B \) is the event "product of digits is zero."
The formula for conditional probability is:

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

In simpler terms, it tells us how likely \( A \) is to happen after we've already seen that \( B \) happened.

Digit Sum

The sum of digits is a straightforward concept where you add up individual digits of a number.Here, we need the tickets whose digits add up to 8.
For example, if we consider the ticket number 35, its digit sum is \( 3 + 5 = 8 \).
An important part of solving our exercise is identifying tickets whose digits sum to 8.When combined with other conditions, this can significantly narrow down possibilities.

Product of Digits Zero

Understanding when the product of the digits of a number equals zero is crucial for this problem.The product of two numbers is zero if at least one of them is zero.
Thus, in a two-digit number, such as a ticket labeled from \( 00 \) to \( 49 \), having a product of digits equal to zero requires that either the tens place or the units place is zero.
For example, in the number 04, since 0 times 4 equals 0, the product is zero.This condition helps us limit the range of possible ticket numbers we need to consider.

Favorable and Possible Outcomes

The concept of favorable and possible outcomes is essential for calculating probability.Here, a favorable outcome is a ticket number that meets both our criteria: the digit sum is 8 and at least one digit (of the ticket) is zero.
In our exercise, tickets such as 08 fit this description.
We identified the total possible outcomes from this range as 14 (numbers whose product of digits is zero). Out of these, only 2 tickets had a digit sum equal to 8.
By dividing favorable outcomes (2) by possible outcomes (14), we obtain our probability, \( \frac{2}{14} = \frac{1}{7} \).
This concept shows how narrowing down possibilities helps effectively determine the probability of specific events.