Given that the events $A$ and $B$ are such that $P(A)=\frac{1}{2}, P(A \cup B)=\frac{3}{5}$ and $P(B)=p .$ Find $p$ if they are independent.
Given that the events $A$ and $B$ are such that $P(A)=\frac{1}{2}, P(A \cup B)=\frac{3}{5}$ and $P(B)=p .$ Find $p$ if they are independent.
When $A$ and $B$ are independent, $P(A \cap B)=P(A) P(B)=\frac{1}{2} p$
It is known that, $P(A \cup B)=P(A)+P(B)-P(A \cap B)$ $\Rightarrow \frac{3}{5}=\frac{1}{2}+p-\frac{1}{2} p$
$\Rightarrow \frac{3}{5}=\frac{1}{2}+\frac{p}{2}$
$\Rightarrow \frac{p}{2}=\frac{3}{5}-\frac{1}{2}=\frac{1}{10}$
$\Rightarrow p=\frac{2}{10}=\frac{1}{5}$
Similar Questions
A fair coin and an unbiased die are tossed. Let $A$ be the event ' head appears on the coin' and $B$ be the event ' $3$ on the die'. Check whether $A$ and $B$ are independent events or not.
A fair coin and an unbiased die are tossed. Let $A$ be the event ' head appears on the coin' and $B$ be the event ' $3$ on the die'. Check whether $A$ and $B$ are independent events or not.
Let $E$ and $F$ be two independent events. The probability that both $E$ and $F$ happens is $\frac{1}{{12}}$ and the probability that neither $E$ nor $F$ happens is $\frac{1}{2},$ then
Let $E$ and $F$ be two independent events. The probability that both $E$ and $F$ happens is $\frac{1}{{12}}$ and the probability that neither $E$ nor $F$ happens is $\frac{1}{2},$ then
- [IIT 1993]
Two students Anil and Ashima appeared in an examination. The probability that Anil will qualify the examination is $0.05$ and that Ashima will qualify the examination is $0.10 .$ The probability that both will qualify the examination is $0.02 .$ Find the probability that Both Anil and Ashima will not qualify the examination.
Two students Anil and Ashima appeared in an examination. The probability that Anil will qualify the examination is $0.05$ and that Ashima will qualify the examination is $0.10 .$ The probability that both will qualify the examination is $0.02 .$ Find the probability that Both Anil and Ashima will not qualify the examination.
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 exactly one of them problem
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 exactly one of them problem
In an entrance test that is graded on the basis of two examinations, the probability of a randomly chosen student passing the first examination is $0.8$ and the probability of passing the second examination is $0.7 .$ The probability of passing at least one of them is $0.95 .$ What is the probability of passing both ?
In an entrance test that is graded on the basis of two examinations, the probability of a randomly chosen student passing the first examination is $0.8$ and the probability of passing the second examination is $0.7 .$ The probability of passing at least one of them is $0.95 .$ What is the probability of passing both ?