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14.Probability
medium
$A , B, C$ try to hit a target simultaneously but independently. Their respective probabilities of hitting targets are $\frac{3}{4},\frac{1}{2},\frac{5}{8}$. The probability that the target is hit by $A$ or $B$ but not by $C$ is
A
$21/64$
B
$7/8$
C
$7/32$
D
$9/64$
(JEE MAIN-2013)
Solution
$P(\text { A or } B \text { but not by } C)=P((A \cup B) \cap \bar{C})$
$=P(A \cup B) \times P(\bar{C})$
$=[\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})] \times \mathrm{P}(\overline{\mathrm{C}})$
$=$ $\left[\frac{3}{4}+\frac{1}{2}-\frac{3}{4} \times \frac{1}{2}\right] \times \frac{3}{8}=\left(\frac{6+4-3}{8}\right) \times \frac{3}{8}=\frac{21}{64}$
Standard 11
Mathematics
