Register

GO Classes DPP | GATE CS, DA | Probability | Bernoulli Random Variables

149 views
1 1 vote

Suppose $X,Y,Z$ are independent binary random variables taking values in ${0,1}$. Let $P(X=0)=p$, $P(Y=0)=q$, and $P(Z=0)=r$. Which expression is equal to $P(X+Y+Z\geq 2\mid X+Y\geq 1)$?

  1. $\frac{(1-p)(1-q)+pq(1-r)}{1-pq}$
     
  2. $\frac{(1-p)(1-q)+(1-r)[p(1-q)+q(1-p)]}{1-pq}$
     
  3. $\frac{(1-p)(1-q)r+(1-r)[p(1-q)+q(1-p)]}{1-pq}$
     
  4. $\frac{1-pqr}{1-pq}$

1 Answer

0 0 votes

We need $P(X+Y+Z\geq 2\mid X+Y\geq 1)$.

Using conditional probability, this is $\frac{P(X+Y+Z\geq 2,\ X+Y\geq 1)}{P(X+Y\geq 1)}$.

Since $X+Y+Z\geq 2$ already forces $X+Y\geq 1$ except in the case $X=0,Y=0,Z=2$, which is impossible because $Z\in{0,1}$, the numerator is just $P(X+Y+Z\geq 2)$.

Now $P(X+Y\geq 1)=1-P(X=0,Y=0)=1-pq$.

The event $X+Y+Z\geq 2$ happens in these cases: $X=1,Y=1,Z=0$, $X=1,Y=0,Z=1$, $X=0,Y=1,Z=1$, or $X=1,Y=1,Z=1$.

So the numerator is $(1-p)(1-q)r+(1-p)q(1-r)+p(1-q)(1-r)+(1-p)(1-q)(1-r)$.

Combine the first and last terms: $(1-p)(1-q)r+(1-p)(1-q)(1-r)=(1-p)(1-q)$.

So the numerator becomes $(1-p)(1-q)+(1-r)[p(1-q)+q(1-p)]$.

$\boxed{\therefore P(X+Y+Z\geq 2\mid X+Y\geq 1)=\frac{(1-p)(1-q)+(1-r)[p(1-q)+q(1-p)]}{1-pq}}$

Voting & Accepting an answer helps improve the community
Answer:
← Previous · GO Classes DPP | GAT...Next → · GO Classes DPP | GAT...
← Previous in category · GO Classes DPP | GAT...Next in category → · GO Classes DPP | GAT...
Position:
Show:

Related questions

1 1 vote
1 1 answer
195
195 views
GO Classes asked Apr 24
195 views
GO Classes DPP | GATE CS, DA | Probability | The Law of Total Probability
Consider three Bernoulli random variables $X$, $Y$, and $Z$, each taking values $0$ or $1$. No independence assumptions are made. Suppose$P(X=1)=\frac{2}{5}$,  $P(Y=1|X=1...
1 1 vote
1 1 answer
173
173 views
GO Classes asked Apr 24
173 views
GO Classes DPP | GATE CS, DA | Probability | Probability Inequalities
Let $A$, $B$, and $C$ be events. Assume every conditional probability that appears is defined. Which statement is NOT necessarily true?If $0<P(B)<1$ and $P(A|B)>P(A)$, th...
1 1 vote
1 1 answer
153
153 views
GO Classes asked Apr 24
153 views
GO Classes DPP | GATE CS, DA | Probability | Chain Rule of Probability
Let $A$, $B$, $C$, and $D$ be events. Assume every conditional probability used below is defined. Without making any independence assumptions, which expression(s) must be...
1 1 vote
1 1 answer
175
175 views
GO Classes asked Apr 24
175 views
GO Classes DPP | GATE CS, DA | Probability | Conditional Probability
A manufacturer tests one electronic chip using four independent inspection gates. The chip passes Gate A, Gate B, Gate C, and Gate D with probabilities $0.96$, $0.90$, $0...