Answer to Question #218236 in Statistics and Probability for Bless

Question

Two events A and B are such that, they are independent and P(A)=x and P(B)=x+0.2 and P(A n B )=0.15. Find;

i) the value of x.

ii) P(A U B) and P( A^c / B^c ), where A^c, B^c are the complements of events A and B respectively.

Answer 1

i) If A and B are independent then $P(A \cap B) = P(A)P(B)$. Then

$x(x + 0.2) = 0.15 \Rightarrow {x^2} + 0.2x - 0.15 = 0$

$D = 0.04 + 0.6 = 0.64$

${x_1} = \frac{{ - 0.2 - 0.8}}{2} = - 0.5$

${x_2} = \frac{{ - 0.2 + 0.8}}{2} = 0.3$

Since $P(A)>0$ then $x=0.3$.

Answer: $x=0.3$

ii)

$P(A) = x = 0.3$

$P(B) = x + 0.2 = 0.5$

Lets find

$P(A \cup B) = P(A) + P(B) - P(A \cap B)=0.3+0.5-0.15=0.65$

Lets find

$P({A^C}) = 1 - P(A) = 0.7$

$P({B^C}) = 1 - P(B) = 0.5$

Then

$P({A^C} \cap {B^C}) = P({A^C})P({B^C}) = 0.7 \cdot 0.5 = 0.35$

So

$P({A^C}\backslash {B^C}) = P({A^C}) - P({A^C} \cap {B^C}) = 0.7 - 0.35 = 0.35$

Answer: $P(A \cup B) =0.65$ , $P({A^C}\backslash {B^C}) = 0.35$