Prawdopodobieństwo warunkowe

[bolek155]

Oblicz \(\displaystyle{ P(A)}\) jeśli \(\displaystyle{ P(B)=2P(B'), P(A\B)= \frac{1}{5}}\) oraz \(\displaystyle{ P(A\B')= \frac{3}{5}}\)

[scyth]

\(\displaystyle{ P(B)=\frac{2}{3} \\
P(B')=\frac{1}{3} \\
P(A \setminus B) = \frac{P(A \cap B)}{P(B)} \ \Rightarrow P(A \cap B) = \frac{2}{3} \cdot \frac{1}{5} = \frac{2}{15} \\
P(A \setminus B') = \frac{P(A \cap B')}{P(B')} \ \Rightarrow P(A \cap B') = \frac{1}{3} \cdot \frac{3}{5} = \frac{3}{15} \\
P(A) = P(A \cap B) + P(A \cap B') = \frac{1}{3}}\)

[bolek155]

czemu w ostatnim równaniu jest 1/3??

[mat_61]

Ponieważ:

\(\displaystyle{ \frac{2}{15} + \frac{3}{15} = \frac{1}{3}}\)