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If [tex]$P(A)=\frac{1}{2}$[/tex], [tex]$P(B)=\frac{1}{5}$[/tex], and [tex]$P(A \text{ and } B)=\frac{1}{10}$[/tex], then A and B are independent events.

True
False

Sagot :

GinnyAnsweridn
To determine whether events [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are independent, we need to verify if the following condition holds:

[tex]\[ P(A \text{ and } B) = P(A) \times P(B) \][/tex]

We are given:
- [tex]\( P(A) = \frac{1}{2} \)[/tex]
- [tex]\( P(B) = \frac{1}{5} \)[/tex]
- [tex]\( P(A \text{ and } B) = \frac{1}{10} \)[/tex]

First, let's calculate [tex]\( P(A) \times P(B) \)[/tex]:

[tex]\[ P(A) \times P(B) = \left( \frac{1}{2} \right) \times \left( \frac{1}{5} \right) = \frac{1}{2} \times \frac{1}{5} = \frac{1}{10} \][/tex]

Now, compare this result with [tex]\( P(A \text{ and } B) \)[/tex]:

[tex]\[ P(A \text{ and } B) = \frac{1}{10} \][/tex]

Since [tex]\( P(A) \times P(B) = \frac{1}{10} \)[/tex] and [tex]\( P(A \text{ and } B) = \frac{1}{10} \)[/tex], we see that:

[tex]\[ P(A \text{ and } B) = P(A) \times P(B) \][/tex]

Therefore, events [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are independent.

Hence, the statement "A and B are independent events." is True.
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