Practice section 6 of 8

$P(A)=0.7$, $P(B\mid A')=0.4$ and $P(A\cap B)=0.28$.

(i) Determine, with a reason, whether A and B are independent. [2]

(ii) Find $P[(A\cap B)\mid(A\cup B)]$. [3]

$P(X\mid Y)=1/2$, $P(Y\mid X)=2/3$ and $P(X\cup Y)=5/6$. Find

(i) $P(X)$, [3]

(ii) $P(X\cup Y')$. [2]

$P(A)=3/5$, $P(B)=3/4$ and $P(A\mid B')=1/2$.

(i) Find $P(A\cap B)$. [2]

(ii) Find $P(A'\mid A\cup B)$. [3]

A third event C has $P(C)=1/2$ and is independent of A.

(iii) Find $P(A'\cap C)$. [1]

(iv) Hence show that $3/40\le P(A'\cap B\cap C)\le1/5$. [2]

A and B are independent events, with $P(A\cup B)=3/4$ and $P(A'\cap B)=2/15$. Find

(i) $P(A)$,

(ii) $P(B)$.

Two fair dice, one white and one red, are thrown. Event A is that the white score is prime; event B is that the sum of the scores is 9.

(i) Find $P(A\mid B)$ and hence determine whether A and B are independent. [3]

(ii) Event C is that the red score is a perfect square. Find $P(B'\cap C')$. [2]

A funfair player draws coloured balls from a box containing five blue and two red balls. To win, the player must pick two red balls consecutively. Each blue ball drawn is replaced and drawing continues; each red ball is not replaced. Calculate the probability of winning. [3]