Union of Sets

Q. Consider four independent trials in which an event $A$ occurs with probability $\frac{1}{3}$. The event $B$ will occur with probability $1$ if the event $A$ occurs at least twice, it can not occur if the event $A$ does not occur and it occurs with a probability $\frac{1}{2}$ if the event $A$ occurs once. If the probability $p$ of the occurrence of event $B$ can be expressed as $\frac{m}{n},$ where $m,n\in N,$ then the least value of $m+n$ is

Q. Every set is a subset of itself.

1. True
2. False

Q.

For the three events A, B and C, P (exactly one of the events A or B occurs) = P(exactly one of the events B or C occurs) = P(exactly one of the events C or A occurs)=p and P(all the three events occur simultaneously)=$p^2$, where 0<p<1/2. Then the probability of at least one of the three events A, B and C occuring is

1. $\frac{3p+2p^2}{2}$
   

2. $\frac{p+3p^2}{4}$
   

3. $\frac{p+3p^2}{2}$
   

4. $\frac{3p+2p^2}{4}$

Q. Two persons $A,B$ speaks truth with probabilities $0.6$ and $0.7$ respectively. The probability that they will say the same thing while describing a single event is:

1. $0.38$
2. $0.94$
3. $0.54$
4. $0.56$

Q.

In a city $20$ percent of the population travels by car, $50$ percent travels by bus and $10$ percent travels by both car and bus. Then the percentage of population travelling by car or bus is

1. $80$ percent

2. $40$ percent

3. $60$ percent

4. $70$ percent