Problem 1 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

Adice is thrown twice and the sum of the numbers appearing is observed to be 6 . What is the conditional probability that the number 4 has appeared at least once?

Step-by-Step Solution

Answer

The conditional probability is \( \frac{2}{5} \).

Step 1: Identify Possible Outcomes

A single die can result in one of 6 outcomes: 1, 2, 3, 4, 5, or 6. When two dice are thrown, there are a total of 6 x 6 = 36 possible outcomes.

Step 2: Calculate Sum of 6 Outcomes

List the outcomes where the sum of the numbers is 6. These outcomes are: (1,5), (2,4), (3,3), (4,2), and (5,1). Thus, there are 5 favorable outcomes when the sum is 6.

Step 3: Identify Favorable Outcomes

Since 4 needs to appear at least once, identify the outcomes from Step 2 that include at least one 4. The favorable pairs here are: (2,4) and (4,2).

Step 4: Calculate Conditional Probability

The probability of event A (4 appearing at least once when the sum is 6) is the ratio of favorable outcomes with a 4 to total outcomes with a sum of 6. Therefore, the conditional probability is \( \frac{2}{5} \).

Key Concepts

Dice Probabilities

Dice probabilities are a key concept when dealing with situations involving dice, such as the exercise about a die thrown twice. When we consider just one die, there are 6 possible outcomes: 1 through 6.
Throwing two dice expands this to a total of 36 possible outcomes, since each die operates independently, and thus, any number appearing on the first die can be coupled with any number on the second. As a result, to determine the probability of any specific outcome or combination of outcomes, we use the basic rule of probability: the number of favorable outcomes divided by the total number of possible outcomes.

• For a single die: 6 outcomes (1-6).
• For two dice: 36 outcomes (each die can be any of 1-6, so 6 possibilities for the first die times 6 for the second).

Each combination of dice yields a distinct outcome, which plays a crucial role in calculating various probabilities associated with specific events.

Probability Theory

Probability theory is the mathematical study of randomness and uncertainty. It provides the tools to model and reason about situations that involve uncertainty, such as throwing a die. Probability tells us how likely it is that a particular event will happen. Events can be seen on a spectrum from certainty (happening 100% of the time) to impossibility (never happening).
In the context of our dice example:

• Each outcome (combination of numbers rolled) has a probability of occurring determined by dividing the number of ways to achieve that outcome by the total possible outcomes.
• Conditional probability is a subpart of probability theory that considers the likelihood of an event, given that another event has already occurred. This is exactly what we solve for when determining the probability of rolling a 4 when the sum is known to be 6.

Understanding these concepts helps us tackle seemingly complex problems, making the outcomes predictable through probability calculations.

Outcomes in Probability

Outcomes in probability refer to the potential results we get from an experiment. In the dice example, an outcome would be the result of a roll of two dice.
To solve the given problem of finding conditional probability, we first identify all possible outcomes that meet the condition of interest, such as the sum being 6. Once identified, the task is to further filter those to include only the outcomes where a 4 appears at least once.

• The possible sums of 6 are: (1,5), (2,4), (3,3), (4,2), and (5,1).
• Favorable outcomes where a 4 appears are: (2,4) and (4,2).

By calculating the ratio of these favorable outcomes to the total outcomes with a sum of 6, we obtain the conditional probability. Understanding outcomes helps us in accurately defining conditions and probabilities, ensuring our predictions are based on solid mathematical foundations.