Three shots are fired independently at a target in succession. The probabilities of a hit in the first shot is , in the second and in the third shot is . In case of one hit, the probability of destroying the target is and in the case of two hits and in the case of three hits 1.0. Find the probability of destroying the target in three shots.,d ij,d ds ckn,d rhu tkrs nwljk rFkk rFkk gSA lgh yxkrk gS] rks u"V gksus gSA blh izdkj nks fy, rFkk rhu fy, 1-0 gSA rks rhu

JEE MainMathematicsToughProbability

Three shots are fired independently at a target in succession.
The probabilities of a hit in the first shot is $\frac{1}{2}$ , in the
second $\frac{2}{3}$ and in the third shot is $\frac{3}{4}$ . In case of
one hit, the probability of destroying the target is $\frac{2}{3}$ and
in the case of two hits $\frac{7}{11}$ and in the case of three hits
1.0. Find the probability of destroying the target in three shots.,d ij,d ds ckn,d rhu tkrs nwljk rFkk
$\frac{1}{2},\frac{2}{3}$ rFkk $\frac{3}{4}$ gSA lgh
yxkrk gS] rks u"V gksus $\frac{2}{3}$ gSA blh izdkj
nks fy, $\frac{7}{11}$ rFkk rhu fy, 1-0 gSA
rks rhu

Show answer and solution

Hit 2 hit
Hit in I^st, II^nd, III^nd

\frac{1}{2}

\frac{2}{3}

\frac{3}{4}

Destroy in I^st II^nd III^rdesa u"V
(1H → Destroy esa u"V) + (2H → Destroy) (2H → Destroy esa u"V)

\left( \frac{1}{2} \cdot \frac{1}{3} \cdot \frac{1}{4} + \frac{1}{2} \cdot \frac{2}{3} \cdot \frac{1}{4} + \frac{1}{2} \cdot \frac{1}{3} \cdot \frac{3}{4} \right) \cdot \frac{2}{3} + \left( \frac{1}{2} \cdot \frac{2}{3} \cdot \frac{1}{4} + \frac{1}{2} \cdot \frac{1}{3} \cdot \frac{3}{4} + \frac{1}{2} \cdot \frac{2}{3} \cdot \frac{3}{4} \right) \cdot \frac{7}{11} + \left( \frac{1}{2} \cdot \frac{2}{3} \cdot \frac{3}{4} \right) \cdot 1

= \frac{6}{24} \cdot \frac{2}{3} + \frac{7}{24} + \frac{6}{24} = \frac{12}{72} + \frac{39}{72} = \frac{51}{72} = \frac{17}{24}

Source: Grass-G / M-21 Mathematics Paper 19 Dec Hindi Eng.docx

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