Q25(ii):
Which of the following arguments are correct and which are not correct? Give reasons for your answer. (ii) If a die is thrown, there are two possible outcomes—an odd number or an even number. Therefore, the probability of getting an odd number is $\frac{1}{2}$.

Solution :

Given: A standard six-faced die is thrown. The possible outcomes are categorized into two groups: odd numbers and even numbers.

To Determine: Whether the argument "the probability of getting an odd number is $\frac{1}{2}$" is correct, and to provide a rigorous mathematical justification.

Visual Representation of the Sample Space:

Step 1: Defining the Sample Space

When a fair six-faced die is thrown, the set of all possible outcomes (Sample Space $S$) is given by:
$S = \{1, 2, 3, 4, 5, 6\}$
The total number of elementary outcomes, denoted by $n(S)$, is $6$.

Step 2: Identifying the Event of Interest

Let $E$ be the event of getting an odd number. The odd numbers present on a standard die are $1, 3,$ and $5$.
Therefore, $E = \{1, 3, 5\}$.
The number of favorable outcomes for event $E$, denoted by $n(E)$, is $3$.

Step 3: Applying the Probability Formula

The theoretical probability of an event $P(E)$ is defined as the ratio of the number of favorable outcomes to the total number of equally likely outcomes:
$P(E) = \frac{n(E)}{n(S)}$
Substituting the values identified in the previous steps:
$P(E) = \frac{3}{6}$

Step 4: Simplifying the Fraction

By dividing both the numerator and the denominator by their greatest common divisor, which is $3$:
$P(E) = \frac{3 \div 3}{6 \div 3} = \frac{1}{2}$

Step 5: Justification of the Argument

The argument states that because there are two categories (odd and even), the probability is $\frac{1}{2}$. While the result $\frac{1}{2}$ is numerically correct, the reasoning is valid only because each of the six outcomes $\{1, 2, 3, 4, 5, 6\}$ is equally likely. Since there are exactly three odd numbers and three even numbers, the probability of selecting an odd number is indeed $\frac{3}{6} = \frac{1}{2}$.

Final Answer: The argument is correct. The probability of getting an odd number is $\frac{1}{2}$ because there are 3 odd outcomes out of 6 equally likely total outcomes.

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Q25(ii): Which of the following arguments are correct and which are not correct? Give reasons for your answer. (ii) If a die is thrown, there are two possible outcomes—an odd number or an even number. Therefore, the probability of getting an odd number is $\frac{1}{2}$.

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Q25(ii):
Which of the following arguments are correct and which are not correct? Give reasons for your answer. (ii) If a die is thrown, there are two possible outcomes—an odd number or an even number. Therefore, the probability of getting an odd number is $\frac{1}{2}$.