When a missile is fired from a ship, the probability that it is intercepted is $\frac{1}{3}$ and the probability that the missile hits the target, given that it is not intercepted, is $\frac{3}{4} .$ If three missiles are fired independently from the ship, then probability that all three hit the target, is:

Option 1 -

$\frac{3}{4}$

Option 2 -

$\frac{1}{27}$

Option 3 -

$\frac{3}{8}$

Option 4 -

$\frac{1}{8}$

2 Views | Posted a month ago

Asked by Shiksha User

1 Answer

Answered by

alok kumar singh | Contributor-Level 10

a month ago

Correct Option - 4

Detailed Solution:

Let

A : Missile hit the target

B : Missile intercepted

P (B) = $\frac{1}{3} P (A / \bar{B}) = \frac{3}{4}$

$P (\bar{B}) = \frac{2}{3}$

$\Rightarrow P (\bar{B} \cap A) = \frac{3}{4} \times \frac{2}{3} = \frac{1}{2}$

Required Probability = $\frac{2}{3} \times \frac{3}{4} \times \frac{2}{3} \times \frac{3}{4} \times \frac{2}{3} \times \frac{3}{4} = \frac{1}{8}$

LetA : Missile hit the targetB : Missile intercepted      P (B) = <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1025" DrawAspect="Content" ObjectID="_1816597409"> </o:OLEObject></xml><! [endif]->  <math> <mrow> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>3</mn> </mrow> </mfrac> <mi>P</mi> <mrow> <mo> (</mo> <mrow> <mi>A</mi> <mo>/</mo> <mover accent="true"> <mi>B</mi> <mo>¯</mo> </mover> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mn>3</mn> </mrow> <mrow> <mn>4</mn> </mrow> </mfrac> </mrow> </math> <math> <mrow> <mi>P</mi> <mrow> <mo> (</mo> <mrow> <mover accent="true"> <mi>B</mi> <mo>¯</mo> </mover> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> </mrow> <mrow> <mn>3</mn> </mrow> </mfrac> </mrow> </math>  <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1026" DrawAspect="Content" ObjectID="_1816597410"> </o:OLEObject></xml><! [endif]-> <math> <mrow> <mo>⇒</mo> <mi>P</mi> <mrow> <mo> (</mo> <mrow> <mover accent="true"> <mi>B</mi> <mo>¯</mo> </mover> <mo>∩</mo> <mi>A</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mn>3</mn> </mrow> <mrow> <mn>4</mn> </mrow> </mfrac> <mo>×</mo> <mfrac> <mrow> <mn>2</mn> </mrow> <mrow> <mn>3</mn> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> </mrow> </math>   <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1027" DrawAspect="Content" ObjectID="_1816597411"> </o:OLEObject></xml><! [endif]->Required Probability = <math> <mrow> <mfrac> <mrow> <mn>2</mn> </mrow> <mrow> <mn>3</mn> </mrow> </mfrac> <mo>×</mo> <mfrac> <mrow> <mn>3</mn> </mrow> <mrow> <mn>4</mn> </mrow> </mfrac> <mo>×</mo> <mfrac> <mrow> <mn>2</mn> </mrow> <mrow> <mn>3</mn> </mrow> </mfrac> <mo>×</mo> <mfrac> <mrow> <mn>3</mn> </mrow> <mrow> <mn>4</mn> </mrow> </mfrac> <mo>×</mo> <mfrac> <mrow> <mn>2</mn> </mrow> <mrow> <mn>3</mn> </mrow> </mfrac> <mo>×</mo> <mfrac> <mrow> <mn>3</mn> </mrow> <mrow> <mn>4</mn> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>8</mn> </mrow> </mfrac> </mrow> </math>  <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1028" DrawAspect="Content" ObjectID="_1816597412"> </o:OLEObject></xml><! [endif]->

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When a missile is fired from a ship, the probability that it is intercepted is 1 3 and the probability that the missile hits the target, given that it is not intercepted, is 3 4 . If three missiles are fired independently from the ship, then probability that all three hit the target, is: