Problem 57 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

A digit is selected from each of the following two sets: \(I=\\{0,1,2,3,4,5,6,7,8,9\\}\) \(I I=\\{0,1,2,3,4,5,6,7,8,9\\}\) The probability that the product of the digits so chosen is positive is (A) \(\frac{4}{5}\) (B) \(\frac{81}{100}\) (C) \(\frac{91}{100}\) (D) none of these

Step-by-Step Solution

Answer

The probability that the product is positive is \(\frac{81}{100}\).

Step 1: Determine Total Outcomes

Each digit is selected independently from sets \(I\) and \(II\), each containing 10 digits. Therefore, the total number of outcomes when selecting one digit from each set is \(10 \times 10 = 100\).

Step 2: Identify Positive Product Conditions

The product of two numbers is positive if either both numbers are positive or both numbers are negative. However, since only digits are selected, all possibilities for sets \(I\) and \(II\) include only non-negative digits from 0 to 9.

Step 3: Exclude Zero as a Factor

To ensure that the product is positive, neither digit can be zero. Thus, we should not select 0 from either set \(I\) or \(II\).

Step 4: Calculate Favorable Outcomes

If 0 is not selected, this means we select from \{1, 2, 3, 4, 5, 6, 7, 8, 9\} for each set, counting 9 options, giving us \(9 \times 9 = 81\) favorable outcomes for a positive product.

Step 5: Calculate Probability

The probability is defined as the ratio of favorable outcomes to the total number of outcomes. Therefore, the probability of both digits being non-zero and hence having a positive product is \( \frac{81}{100} \).

Key Concepts

Positive Product

To determine when the product of two numbers is positive, we need to look at the signs of the numbers involved. If both numbers are positive, their product will also be positive. Likewise, if both numbers are negative, their product is positive too. However, our sets only include non-negative digits (0 to 9), eliminating the possibility of negative numbers. Hence, in this context, to have a positive product, both numbers must simply be non-zero. Selecting any number except zero from each set ensures a positive product. In our scenario, neither digit selected from set \(I\) nor set \(II\) can be zero for the product to stay positive. This fundamental understanding of when a product is positive is critical in solving the given problem effectively.

Favorable Outcomes

In probability, favorable outcomes are those that represent the event we are interested in. For this problem, we are interested in outcomes where the product of selected digits from sets \(I\) and \(II\) is positive.To calculate the number of favorable outcomes, we ensure neither digit chosen is zero.

• Both sets have numbers from \(1\) to \(9\) as options, excluding zero.
• Each set therefore provides us with 9 potential positive number choices.

Given that both sets are independent of each other, the total number of favorable outcomes comes from multiplying the options of each set: \(9 \times 9 = 81\). This shows there are 81 different ways to select a pair of digits that have a positive product.

Total Outcomes

The total number of outcomes is found by considering how many different ways digits can be selected from both sets. Set \(I\) contains 10 digits, as does set \(II\).

• We select one digit from set \(I\) and one from set \(II\)
• This is done independently, giving us the combination calculation of \(10 \times 10\).

Calculating these combinations results in 100 total outcomes. Understanding this is important because the denominator in probability calculations often comes from the total number of possible outcomes. These 100 possibilities provide the backdrop against which we assess the favorable outcomes, allowing us to calculate probabilities accurately. In this exercise, it is through this structure that we place the 81 favorable outcomes we calculated earlier, within the context of the total 100 outcomes.