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Probability — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsProbability2 Marks
MCQ
A problem in mathematics is given to 4 students whose chances of solving individually are $\frac{1}{2}, \frac{1}{3}, \frac{1}{4}$ and $\frac{1}{5}$. Then probability that the problem will be solved at least by one student is
  • A
    $\frac{2}{3}$
  • B
    $\frac{3}{5}$
  • ✓
    $\frac{4}{5}$
  • D
    $\frac{3}{4}$
✓

Answer

Correct option: C.
$\frac{4}{5}$
(C)
Let $A , B , C$ and D be the events that the problem will be solved by $1^{\text {st }}, 2^{\text {nd }}, 3^{\text {rd }}$ and $4^{\text {th }}$ students respectively.
Probability that no student solve the problem
$= P (\overline{ A } \cap \overline{ B } \cap \overline{ C } \cap \overline{ D })$
$=\left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right)\left(1-\frac{1}{4}\right)\left(1-\frac{1}{5}\right)=\frac{1}{2} \times \frac{2}{3} \times \frac{3}{4} \times \frac{4}{5}=\frac{1}{5}$
$\ldots[$ As $\overline{ A }, \overline{ B }, \overline{ C }, \overline{ D }$ are independent events $]$
⇒ Probability that the problem will be solved by at least one student $= P ( A \cup B \cup C \cup D)$
$=1- P (\overline{ A } \cap \overline{ B } \cap \overline{ C } \cap \overline{ D })$
$=1-\frac{1}{5}=\frac{4}{5}$

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