Quiz questions

Events Decide which of the following statements are true?

(i) If $A$ and $B$ are exclusive events then so are $A^c$ and $B^c$.

(ii) $A\backslash B=B\backslash A$.

(iii) $A\cap B^c=A\backslash B$.

(A) TTT, (B) FFT, (C) FTF, (D) TFF, (E) FFF.

Vets An experiment is carried out where the next animal to be brought into a vet is observed. Suppose the sample space $\mathrm{\Omega }$ contains three outcomes, $\mathrm{\Omega }=\{\text{ cat, dog, rabbit }\}$. Assuming $\mathrm{P}$ satisfies axiom 3, decide whether the following are valid probabilities.

(i) $\mathrm{P}(\{\text{cat}\})=\mathrm{P}(\{\text{dog}\})=\mathrm{P}(\{\text{rabbit}\})=1/4$.

(ii) $\mathrm{P}(\{\text{cat}\})=1/6$, $\mathrm{P}(\{\text{dog}\})=1/3$, $\mathrm{P}(\{\text{rabbit}\})=1/2$.

(iii) $\mathrm{P}(\{\text{cat}\})=0.3$, $\mathrm{P}(\{\text{dog}\})=-0.1$, $\mathrm{P}(\{\text{rabbit}\})=0.8$.

(A) TTT, (B) TTF, (C) FTF, (D) FTT, (E) FFF.

Conditional probabilities Which of the following statements are true?

(a) If $A$ and $B$ are exclusive events then $\mathrm{P}(A|B)=0$.

(b) If $A\subset B$ are events then $\mathrm{P}(A|B)=\mathrm{P}(A)/\mathrm{P}(B)$.

(c) If $A\subset B$ are events then $\mathrm{P}(B|A)=1$.

(A) TTT, (B) TTF, (C) TFT, (D) FTT, (E) FFF.

Vectorised command Assuming that x and y are vectors of the same length, consider the following code:

m <- rep(0,length(x))

for (i in 1:length(x)) {

m[i] <- max(x[i],y[i])

}

What operation will produce the same vector m without the need for a for loop?

1. (A)

   m <- max(x,y)

2. (B)

   m <- x*(x >=y) + y*(x<y)

3. (C)

   m <- y*(x >=y) + x*(x<y)

4. (D)

   m <- x*(max(x,y)==x) + y*(max(x,y)==y)

5. (E)

   m <- x*(max(x,y)==x) + y*(max(x,y)!=x)

Recursion relationship II A sequence $b_n$ is defined by the inhomogeneous recursion relationship

where $b_1=1,b_2=1/2$.

Adapt the recursion function to compute the first $N$ values of an inhomogeneous recursion relationship and hence find the value of $k$ for which $b_{100}=0$ (or very close to 0).

(A) $k=0.3319$, (B) $k=-0.1323$,  (C) $k=1.006$,  (D) $k=-1.006$,  (E) $k=0.4561$