Problem 25 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

Two dice are thrown. What is the probability that the sum of the faces equals or exceeds \(10 ?\) (a) \(1 / 12\) (b) \(1 / 4\) (c) \(1 / 3\) (d) \(1 / 6\)

Step-by-Step Solution

Answer

The probability that the sum equals or exceeds 10 is \(\frac{1}{6}\), option (d).

Step 1: List Possible Outcomes

Consider that each die has 6 faces. When two dice are thrown, each die has 6 options leading to a total of \(6 \times 6 = 36\) possible outcomes. Each outcome is an ordered pair \((a, b)\) where \(a\) and \(b\) are the results of the first and second dice, respectively. Examples include \((1,1), (1,2), ... , (6,6)\).

Step 2: Determine Successful Outcomes

Calculate the pairs where the sum \(a + b\) is 10 or more: - Sum equals 10: \((4,6), (5,5), (6,4)\)- Sum equals 11: \((5,6), (6,5)\)- Sum equals 12: \((6,6)\)Thus, there are 6 successful outcomes where the sum equals or exceeds 10.

Step 3: Calculate Probability

The probability of an event is given by the ratio of successful outcomes to total possible outcomes. In this case, the probability \(P\) is \(\frac{\text{Successful Outcomes}}{\text{Total Possible Outcomes}} = \frac{6}{36} = \frac{1}{6}\).

Key Concepts

Probability of Dice

When we talk about dice in probability, we're referring to those familiar six-sided cubes. Each side has a number from 1 to 6. When two dice are rolled, each die independently shows one of these numbers. The goal is often to determine the probability of certain results, like the sum of the die faces equaling or exceeding a certain value. For example, if you aim to find the probability that the total of two dice equals or exceeds 10, you consider all possible outcomes, determine which satisfy your condition (sum equals or exceeds 10), and then compare that to the total possible outcomes.

Outcomes in Probability

The term "outcomes" in probability refers to the possible results of an experiment. For two dice, each with 6 faces, each die can roll a 1 through a 6. This gives us:

• 36 possible combinations or outcomes, calculated using the multiplication rule \(6 \times 6 = 36\).
• An ordered pair \( (a, b) \) represents an outcome where \(a\) is the result from the first die and \(b\) is the result from the second die.

Examples of possible outcomes include (1,1), (1,2), ..., up to (6,6). Identifying outcomes is the first step to solving any probability problem because you need to know all possibilities before working towards finding the ones that meet your criteria.

Combinatorics

Combinatorics is a field of mathematics focused on counting, arrangement, and combination of elements. When dealing with two dice, combinatorics helps us identify and count the possible outcomes. By learning to count how many of each specific outcome there are, like how many combinations result in a sum of 10 or more, we can solve probability problems.

• In the problem at hand, successful outcomes could be combinations such as (4,6), (5,5), and (6,6).
• The formulas and principles of combinatorics allow us to determine these scenarios efficiently.
• This approach is especially useful in more complex problems where listing each outcome could be impractical.

With combinatorics, you're essentially building a toolkit for probability, allowing you to systematically work through problems and derive probabilities for various scenarios.