Events $E$ and $F$ are such that $P ( $ not $E$ not $F )=0.25,$ State whether $E$ and $F$ are mutually exclusive.
Events $E$ and $F$ are such that $P ( $ not $E$ not $F )=0.25,$ State whether $E$ and $F$ are mutually exclusive.
It is given that $P$ (not $E$ or not $F$ ) $=0.25$
i.e., $P \left( E ^{\prime} \cap F ^{\prime}\right)=0.25$
$\Rightarrow P ( E \cap F )^{\prime} =0.25$ $[ E^{\prime} \cup F^{\prime} =( E \cap F )^{\prime}]$
Now, $P ( E \cap F )=1- P ( E \cap F )^{\prime}$
$\Rightarrow P ( E \cap F )=1-0.25$
$\Rightarrow P ( E \cap F )=0.75 \neq 0$
$\Rightarrow E \cap F \neq \phi$
Thus, $E$ and $F$ are not mutually exclusive.
Similar Questions
If $A$ and $B$ are two events such that $P(A) = \frac{1}{2}$ and $P(B) = \frac{2}{3},$ then
If $A$ and $B$ are two events such that $P(A) = \frac{1}{2}$ and $P(B) = \frac{2}{3},$ then
An unbiased coin is tossed. If the result is a head, a pair of unbiased dice is rolled and the number obtained by adding the numbers on the two faces is noted. If the result is a tail, a card from a well shuffled pack of eleven cards numbered $2, 3, 4,.......,12$ is picked and the number on the card is noted. The probability that the noted number is either $7$ or $8$, is
An unbiased coin is tossed. If the result is a head, a pair of unbiased dice is rolled and the number obtained by adding the numbers on the two faces is noted. If the result is a tail, a card from a well shuffled pack of eleven cards numbered $2, 3, 4,.......,12$ is picked and the number on the card is noted. The probability that the noted number is either $7$ or $8$, is
- [IIT 1994]
One card is drawn from a pack of $52$ cards. The probability that it is a queen or heart is
One card is drawn from a pack of $52$ cards. The probability that it is a queen or heart is
Probability of solving specific problem independently by $A$ and $B$ are $\frac{1}{2}$ and $\frac{1}{3}$ respectively. If both try to solve the problem independently, find the probability that the problem is solved.
Probability of solving specific problem independently by $A$ and $B$ are $\frac{1}{2}$ and $\frac{1}{3}$ respectively. If both try to solve the problem independently, find the probability that the problem is solved.
Consider an experiment of tossing a coin repeatedly until the outcomes of two consecutive tosses are same. If the probability of a random toss resulting in head is $\frac{1}{3}$, then the probability that the experiment stops with head is.
Consider an experiment of tossing a coin repeatedly until the outcomes of two consecutive tosses are same. If the probability of a random toss resulting in head is $\frac{1}{3}$, then the probability that the experiment stops with head is.
- [IIT 2023]