Math  /  Data & Statistics

Question1.21. Ngjarjet AA the BB jané té tilla qè: P(A)=1/2,P(B)=1/3,P(AB)=1/4P(A)=1 / 2, P(B)=1 / 3, P(A B)=1 / 4. Njehesoni: P(A+B),P(A/B),P(A/Bˉ),P(AB),P(Aˉ/Bˉ)P(A+B), P(A / B), P(A / \bar{B}), P(\overline{A B}), P(\bar{A} / \bar{B}).

Studdy Solution
Use the formula for conditional probability with the complements:
P(AˉBˉ)=P(AˉBˉ)P(Bˉ) P(\bar{A} \mid \bar{B}) = \frac{P(\bar{A} \cap \bar{B})}{P(\bar{B})}
First, find P(Aˉ) P(\bar{A}) :
P(Aˉ)=1P(A)=112=12 P(\bar{A}) = 1 - P(A) = 1 - \frac{1}{2} = \frac{1}{2}
Next, find P(AˉBˉ) P(\bar{A} \cap \bar{B}) :
P(AˉBˉ)=P(Aˉ)+P(Bˉ)P(AˉBˉ) P(\bar{A} \cap \bar{B}) = P(\bar{A}) + P(\bar{B}) - P(\bar{A} \cup \bar{B})
Using De Morgan's laws:
P(AˉBˉ)=1P(AB)=34 P(\bar{A} \cup \bar{B}) = 1 - P(A \cap B) = \frac{3}{4}
Now, calculate:
P(AˉBˉ)=12+2334 P(\bar{A} \cap \bar{B}) = \frac{1}{2} + \frac{2}{3} - \frac{3}{4}
Find a common denominator and calculate:
P(AˉBˉ)=612+812912=512 P(\bar{A} \cap \bar{B}) = \frac{6}{12} + \frac{8}{12} - \frac{9}{12} = \frac{5}{12}
Finally, calculate P(AˉBˉ) P(\bar{A} \mid \bar{B}) :
P(AˉBˉ)=51223=512×32=58 P(\bar{A} \mid \bar{B}) = \frac{\frac{5}{12}}{\frac{2}{3}} = \frac{5}{12} \times \frac{3}{2} = \frac{5}{8}
The probabilities are:
P(AB)=712 P(A \cup B) = \frac{7}{12} P(AB)=34 P(A \mid B) = \frac{3}{4} P(ABˉ)=38 P(A \mid \bar{B}) = \frac{3}{8} P(AB)=34 P(\overline{A \cap B}) = \frac{3}{4} P(AˉBˉ)=58 P(\bar{A} \mid \bar{B}) = \frac{5}{8}

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