Problem 9 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

\(A\) and \(B\) throw a dice. The probability that \(A\) 's throw is not greater than B's is (A) \(\frac{5}{12}\) (B) \(\frac{7}{12}\) (C) \(\frac{1}{6}\) (D) \(\frac{1}{2}\)

Step-by-Step Solution

Answer

The probability is \(\frac{7}{12}\), which is option (B).

Step 1: Understand the Problem

Both players, A and B, throw a single six-sided die. We are looking to find the probability that the result of A's throw is not greater than the result of B's throw.

Step 2: Determine Sample Space

Each player can roll a die, resulting in one of 6 outcomes (1 through 6). Therefore, the total number of possible outcomes when both players throw their dice is \(6 \times 6 = 36\).

Step 3: Identify Favorable Outcomes

A's throw is not greater than B's when either A equals B or A is less than B. Enumerate these outcomes: - A = B: Possible with combinations (1,1), (2,2), (3,3), (4,4), (5,5), (6,6). - A < B: For each value of A, count outcomes where B is greater. - A=1: B=2,3,4,5,6 (5 outcomes) - A=2: B=3,4,5,6 (4 outcomes) - A=3: B=4,5,6 (3 outcomes) - A=4: B=5,6 (2 outcomes) - A=5: B=6 (1 outcome) Total favorable outcomes are 6 + 5 + 4 + 3 + 2 + 1 = 21.

Step 4: Calculate the Probability

The probability is the number of favorable outcomes divided by the total number of possible outcomes. \[P(A \leq B) = \frac{21}{36} = \frac{7}{12}\]

Step 5: Verify if Answer Matches Given Options

The calculated probability \(\frac{7}{12}\) matches option B.

Key Concepts

Dice Probability

When you roll a die, you're exploring the exciting world of probability. Probability simply refers to how likely an event is to happen. In the context of dice, it is about predicting the chances of rolling a particular number.
Each side of a six-sided die has an equal chance of landing face up.
This probability is \ \( \frac{1}{6} \ \), since there are six sides.
When both players throw a dice, we calculate the likelihood of different outcomes occurring by evaluating the probability of each specific event.

Having a good grasp of dice probability is crucial for games and other situations involving chance. It is essential to understand that each roll is independent of the others. This means the outcome of one roll does not affect another roll.
In our exercise, we use probability to determine how often Player A's dice roll is less than or equal to Player B's.
This involves calculating all possible outcomes and identifying which ones satisfy this condition.

Sample Space

In probability, the concept of a sample space is pivotal.
The sample space of an experiment is the set of all possible outcomes. For dice, when you roll a single six-sided die, the sample space is \( \{1, 2, 3, 4, 5, 6\} \).
However, when both A and B roll their dice in the exercise, the scenario becomes a bit more complex.

The sample space for two dice rolls should consider every possible combination of outcomes from both dice.

• Think of it as a grid, with 6 possibilities for A and 6 possibilities for B, totaling 36 outcomes.
• Examples of these outcomes include \((1, 1), (1, 2), ..., (6, 6)\).
• This comprehensive view allows us to count how many possible interactions A's and B's dice rolls can have.

Recognizing the sample space correctly helps in identifying favorable outcomes accurately, which is vital for solving any probability problem.

Favorable Outcomes

In probability exercises, once you have defined your sample space, the next step is to identify the favorable outcomes.
These are the outcomes that satisfy the condition of the problem statement. In our dice exercise, a favorable outcome occurs when Player A's roll is not greater than Player B's.

The exercise solution involved counting these specific outcomes.

• For when A is equal to B, there are 6 combinations: \((1,1), (2,2), ..., (6,6)\).
• For A being less than B, the combinations include several sequences depending on the number A rolls.
• Combining all possible favorable scenarios results in 21 out of 36 unique outcomes.

The ratio of favorable outcomes to the total number of outcomes gives the probability that A's roll is not greater than B's.
Understanding how to count and use favorable outcomes is key in accurately determining probabilities in any dice game scenario.