To solve this problem, we need to use the property of independent events. For independent events [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[
P(A \text{ and } B) = P(A) \cdot P(B)
\][/tex]
We are given:
- [tex]\( P(A) = \frac{2}{5} \)[/tex]
- [tex]\( P(A \text{ and } B) = \frac{1}{10} \)[/tex]
We need to determine [tex]\( P(B) \)[/tex].
Using the formula for the probability of independent events occurring together:
[tex]\[
\frac{1}{10} = \frac{2}{5} \cdot P(B)
\][/tex]
To isolate [tex]\( P(B) \)[/tex], we divide both sides of the equation by [tex]\( \frac{2}{5} \)[/tex]:
[tex]\[
P(B) = \frac{\frac{1}{10}}{\frac{2}{5}}
\][/tex]
Dividing fractions is equivalent to multiplying by the reciprocal of the divisor:
[tex]\[
P(B) = \frac{1}{10} \cdot \frac{5}{2}
\][/tex]
Now, we perform the multiplication:
[tex]\[
P(B) = \frac{1 \cdot 5}{10 \cdot 2} = \frac{5}{20} = \frac{1}{4}
\][/tex]
So, the probability of event [tex]\( B \)[/tex] occurring is [tex]\( \frac{1}{4} \)[/tex].
Thus, the value of [tex]\( P(B) \)[/tex] is:
[tex]\[
\boxed{\frac{1}{4}}
\][/tex]
