Problem 98 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Four tickets marked \(00,01,10,11\), respectively are placed in a bag. A ticket is drawn at random five times, being replaced each time. The probability that the sum of the numbers on tickets thus drawn is 23 is (A) \(\frac{25}{256}\) (B) \(\frac{100}{256}\) (C) \(\frac{231}{256}\) (D) none of these

Step-by-Step Solution

Answer

The probability is 0.

Step 1: Understanding the Problem

We have a set of four tickets marked as binary numbers: \( 00, 01, 10, 11 \). Each time a ticket is drawn, there are four possible outcomes. And this is done five times with replacement. We need to find the probability that the sum of the binary numbers is 23.

Step 2: Converting Binary to Decimal

Convert each binary number to decimal: \( 00 \) is \( 0 \), \( 01 \) is \( 1 \), \( 10 \) is \( 2 \), and \( 11 \) is \( 3 \). We are summing these values after each draw.

Step 3: Total Number of Outcomes

Since there are four possible outcomes for each draw and the ticket is drawn five times, the total number of possible outcomes is \( 4^5 = 1024 \).

Step 4: Possible Combination Sum to 23

Calculate combinations of the draws that sum to 23. The maximum sum, if we pick \( 11 \) each time, is \( 3 \times 5 = 15 \). Since the maximum possible sum with the numbers we have is 15, it's impossible to reach 23.

Step 5: Conclusion on Possibility

Since the maximum sum of 15 (drawing \( 11 \) each time for all five draws) is less than 23, getting a sum of 23 is impossible with the available numbers.

Step 6: Determining the Probability

Since it's impossible to have a sum of 23, the probability is 0.

Key Concepts

Binary Numbers Unveiled

Binary numbers are a fundamental part of computer science and mathematics. They use only two digits: 0 and 1, which is why they are called 'binary.'
Each position in a binary number represents a power of 2, starting with 2 raised to the power of 0 on the furthest right. For example, the binary number 1101 is calculated as follows:

• The rightmost digit, 1, represents 2^0 which is 1.
• The next digit to the left, 0, represents 2^1 which is 2, but because it's 0, it adds 0 to the total.
• The next digit is 1, representing 2^2, which is 4.
• The leftmost digit, another 1, represents 2^3, which is 8.

Adding these values together, 1 + 0 + 4 + 8, gives the decimal equivalent of 13. Understanding this conversion is crucial to solving problems that involve switching between binary and decimal systems.

Decimal Conversion Made Easy

Decimal conversion is the process of translating numbers from the binary system into the decimal system that most people use daily. Decimal is a base-10 system, meaning it uses ten digits: 0 through 9.
To convert a binary number to decimal:

• Write down the binary number.
• Starting from the right, multiply each binary digit by the power of 2 that corresponds to its position.
• Sum all these values to get the decimal equivalent.

For instance, the binary number 1010 translates to decimal as follows:
- The rightmost 0 represents 0 times 2^0, which is 0.
- The next 1 represents 1 times 2^1, which equals 2.
- The following 0 represents 0 times 2^2, adding 0.
- The leftmost 1 represents 1 times 2^3, which is 8.
By adding them, 0 + 2 + 0 + 8, the decimal number is 10. Converting accurately is imperative for calculating sums when dealing with binary numbers.

Exploring Combinatorics

Combinatorics is a field in mathematics that deals with counting, arranging, and finding patterns in sets of elements. It's particularly useful in probability when determining possible outcomes.
When you draw items with replacement, like in our exercise, each draw is independent. Hence, the formulas for combinations or permutations need to adapt to these conditions.

• For a single event with multiple outcomes, such as drawing a ticket, the total number of possible outcomes is raised to the power of the number of events (repeated draws).
• In the exercise example, there are four tickets and five draws, resulting in 4^5, or 1024, possible outcomes.
• Combinatorics also allows us to calculate the probability of certain combinations, but only when they make sense, such as sums within possible number ranges.

In situations where a particular sum is impossible, as in trying to reach a sum of 23 with our ticket values, combinatorics helps us arrive quickly at the conclusion that the probability is zero.