Problem 21 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

Two dice are thrown. The probability that the sum of numbers appearing is more than 10 is (a) \(1 / 18\) (b) \(1 / 12\) (c) \(1 / 6\) (d) None of these

Step-by-Step Solution

Answer

The probability that the sum is more than 10 is \(\frac{1}{12}\), so the answer is (b).

Step 1: Understand Total Outcomes

When two six-sided dice are thrown, each die has 6 faces. Therefore, the total number of possible outcomes is \(6 \times 6 = 36\).

Step 2: Sum Conditions

We are interested in the outcomes where the sum of the numbers is more than 10. The possible sums greater than 10 are 11 and 12.

Step 3: Find Favorable Outcomes for Sum 11

To find outcomes where the sum is 11, consider the pairs: \((5, 6)\) and \((6, 5)\). Thus, there are 2 favorable outcomes for a sum of 11.

Step 4: Find Favorable Outcomes for Sum 12

To achieve a sum of 12, the only possible pair is \((6, 6)\). Thus, there is 1 favorable outcome for a sum of 12.

Step 5: Calculate Total Favorable Outcomes

Adding the favorable outcomes for sums 11 and 12 gives a total of \(2 + 1 = 3\) favorable outcomes.

Step 6: Calculate Probability

The probability is calculated by dividing the number of favorable outcomes by the total number of outcomes. Therefore, the probability that the sum is more than 10 is \(\frac{3}{36} = \frac{1}{12}\).

Key Concepts

Dice Probabilities

Dice probabilities help us to determine the likelihood of a particular outcome when rolling dice. When you roll two dice, each die has 6 faces with numbers ranging from 1 to 6. Thus, for a single die, the probability of any specific number is one in six or \(\frac{1}{6}\). For two dice, the probabilities evolve because there are 36 possible combinations.
Understanding dice probabilities involves knowing:

• The total number of possible outcomes (which is 36, from \(6 \times 6\))
• The desired sum or outcome you are calculating probability for
• The number of favorable combinations for that sum or result

In dice games and problems, the essence is to comprehend how each roll or combination influences the probability, allowing predictions on results.

Outcome Analysis

Outcome analysis refers to the process of examining all possible results of throwing dice to understand how often a particular event can occur. Whenever we analyze a problem involving dice, we list all potential outcomes and see which ones meet our conditions.

Example: Rolling Two Dice

When two dice are thrown, they can produce numerous total scores ranging from 2 to 12. To conduct an outcome analysis:

• Consider all pairs, such as (1,1), (1,2), ..., (6,6).
• Calculate the sum for each combination.
• Count the number of combinations that meet the criteria, like achieving a sum greater than 10.

Thus, outcome analysis enables us to predict and strategize effectively by clearly knowing which outcomes are possible.

Sum of Dice

The sum of dice refers to the combined total when the numbers on the top faces of multiple dice are added together. This concept is central to games and probability problems involving dice. Understanding which sums can arise and how frequently they appear involves a straightforward method.

Key Points: Understanding Dice Sums

• A single die can result in numbers from 1 to 6.
• When two dice are rolled, possible sums range from 2 (1+1) to 12 (6+6).
• Each sum between 2 and 12 has a different probability based on how many combinations can achieve it.

By observing the frequency of possible sums, players or problem solvers can anticipate or calculate outcomes with a fair degree of accuracy.

Favorable Outcomes

Favorable outcomes are the specific results that align with the conditions set by a problem. In probability, determining these outcomes is crucial for calculating how likely an event is to occur.

Example of Favorable Outcomes

Consider rolling two dice and wanting to find sums greater than 10. You identify and count the rolls yielding such sums:

• For a sum of 11, possible rolls: (5,6), (6,5).
• For a sum of 12, possible roll: (6,6).

There are three favorable outcomes. Once identified, these outcomes allow us to calculate the desired probability, giving the chance of these sums occurring as a fraction of all possible results (3 out of 36). Recognizing favorable outcomes helps to simplify complex probability queries into manageable calculations.