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Step-by-Step Solution
Step 1: Identify the Total Number of Possible Outcomes
When a fair die is thrown once, the set of all possible outcomes (the sample space) is
{1, 2, 3, 4, 5, 6}, which has 6 outcomes in total.
Thus, the total number of outcomes is 6.
Step 2: Define the Events A and B
• Event A = "Number obtained is greater than 3"
Therefore, $A = \{4, 5, 6\}$.
Hence, $P(A) = \frac{3}{6} = \frac{1}{2}.$
• Event B = "Number obtained is less than 5"
Therefore, $B = \{1, 2, 3, 4\}.$
Hence, $P(B) = \frac{4}{6} = \frac{2}{3}.$
Step 3: Find the Intersection of A and B
The intersection $A \cap B$ consists of outcomes which are in both A and B simultaneously.
$A \cap B = \{4\}.$
So, $P(A \cap B) = \frac{1}{6}.$
Step 4: Apply the Formula for the Probability of the Union of Two Events
We use the formula:
$P(A \cup B) = P(A) + P(B) - P(A \cap B).$
Substituting the values:
$P(A \cup B) = \frac{1}{2} + \frac{2}{3} - \frac{1}{6}.$
Calculate step by step:
• $\frac{1}{2} = \frac{3}{6}$,
• $\frac{2}{3} = \frac{4}{6}$,
• So, $P(A \cup B) = \frac{3}{6} + \frac{4}{6} - \frac{1}{6} = \frac{6}{6} = 1.$
Step 5: State the Final Answer
Therefore, $P(A \cup B) = 1.$
Hence, the correct answer is 1.
