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.

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Probability of solving the problem by $\mathrm{A}, \mathrm{P}(\mathrm{A})=\frac{1}{2}$

Probability of solving the problem by $\mathrm{B}, \mathrm{P}(\mathrm{B})=\frac{1}{3}$

since the problem is solved independently by $A$ and $B$,

$\therefore $ $\mathrm{P}(\mathrm{AB})=\mathrm{P}(\mathrm{A}) \cdot \mathrm{P}(\mathrm{B})=\frac{1}{2} \times \frac{1}{3}=\frac{1}{6}$

$P(A^{\prime})=1-P(A)=1-\frac{1}{2}=\frac{1}{2}$

$P(B^{\prime})=1-P(B)=1-\frac{1}{3}=\frac{2}{3}$

Probability that the problem is solved $=\mathrm{P}(\mathrm{A} \cup \mathrm{B})$

$=\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{AB})$

$=\frac{1}{2}+\frac{1}{3}-\frac{1}{6}$

$=\frac{4}{6}$

$=\frac{2}{3}$

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  • [JEE MAIN 2020]

Let ${E_1},{E_2},{E_3}$ be three arbitrary events of a sample space $S$. Consider the following statements which of the following statements are correct

One card is drawn at random from a well shuffled deck of $52$ cards. In which of the following cases are the events $\mathrm{E}$ and $\mathrm{F}$ independent ?

$E:$ ' the card drawn is a king and queen '

$F:$  ' the card drawn is a queen or jack '