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Sagot :
Answer:
c
Step-by-step explanation:
because we have already learned that lesson and that is the answer
I hope I can help you
Answer:
Basic Setup
A die has 6 faces, numbered 1 through 6. The sample space \( S \) when a die is rolled is:
[tex][ S = \{ 1, 2, 3, 4, 5, 6 \} \][/tex]
Part (a): Probability that the number is even or a multiple of 3
Define the events:
- ( E ): The number is even.
- ( M ): The number is a multiple of 3.
First, identify the members of each set:
[tex]E = \{ 2, 4, 6 \}[/tex]
[tex]M = \{ 3, 6 \}[/tex]
The union of the two events
[tex] \( E \cup M \):[/tex]
[tex]E \cup M = \{ 2, 3, 4, 6 \}[/tex]
[tex]{\[ P(E \cup M) = P(E) + P(M) - P(E \cap M) \]}[/tex]
Calculate individual probabilities:
[tex]P(E) = \frac{|E|}{|S|} = \frac{3}{6} = \frac{1}{2}[/tex]
[tex]P(M) = \frac{|M|}{|S|} = \frac{2}{6} = \frac{1}{3}[/tex]
[tex]E \cap M = \{ 6 \} \Rightarrow P(E \cap M) = \frac{1}{6}[/tex]
Thus,
[tex]{P(E \cup M) = \frac{1}{2} + \frac{1}{3} - \frac{1}{6} = \frac{3}{6} + \frac{2}{6} - \frac{1}{6} = \frac{4}{6} = \frac{2}{3}}[/tex]
Part (b): Probability that the number is a multiple of 2 or multiple of 3
Define the events:
- ( A ): The number is a multiple of 2.
- ( B ): The number is a multiple of 3.
Identify the members of each set:
[tex]A = \{ 2, 4, 6 \}[/tex]
[tex]B = \{ 3, 6 \}[/tex]
The union of the two events
[tex]( A \cup B \):[/tex]
[tex]A \cup B = \{ 2, 3, 4, 6 \}[/tex]
The probability of the union:
[tex]{P(A \cup B) = P(A) + P(B) - P(A \cap B)}[/tex]
Calculate individual probabilities:
[tex]P(A) = \frac{|A|}{|S|} = \frac{3}{6} = \frac{1}{2}[/tex]
[tex]P(B) = \frac{|B|}{|S|} = \frac{2}{6} = \frac{1}{3}[/tex]
[tex]A \cap B = \{ 6 \} \Rightarrow P(A \cap B) = \frac{1}{6}[/tex]
Thus,
[tex]{P(A \cup B) = \frac{1}{2} + \frac{1}{3} - \frac{1}{6} = \frac{3}{6} + \frac{2}{6} - \frac{1}{6} = \frac{4}{6} = \frac{2}{3}}[/tex]
Part (c): Probability that the number is an odd number or a multiple of 2
Define the events:
- ( O ): The number is odd.
- ( D ): The number is a multiple of 2 (i.e., the number is even).
Identify the members of each set:
[tex] O = \{ 1, 3, 5 \}[/tex]
[tex]D = \{ 2, 4, 6 \}[/tex]
The union of the two events
[tex]\( O \cup D \):[/tex]
[tex]O \cup D = \{ 1, 2, 3, 4, 5, 6 \}[/tex]
The probability of the union:
[tex]{P(O \cup D) = P(O) + P(D) - P(O \cap D)}[/tex]
Calculate individual probabilities:
[tex]P(O) = \frac{|O|}{|S|} = \frac{3}{6} = \frac{1}{2}[/tex]
[tex]P(D) = \frac{|D|}{|S|} = \frac{3}{6} = \frac{1}{2}[/tex]
[tex]\[ O \cap D[/tex]
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