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Sagot :
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]
[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]
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