JEE Main · 2020 · 05 Sep 2020 · Shift 2 · Probability

In a bombing attack, there is

50%

chance that a bomb will hit the target. At least two independent hits are required to destroy the target completely. Then the minimum number of bombs, that must be dropped to ensure that there is at least

99%

chance of completely destroying the target, is ____ .

Answer: 11

Solution

Each bomb hits with probability

\frac{1}{2}

, and at least two hits are needed. The chance of fewer than two hits among

n

bombs is

\frac{1}{2 ^{n}} + \frac{n}{2 ^{n}} = \frac{n + 1}{2 ^{n}}

, so the target is destroyed with probability

1 - \frac{n + 1}{2 ^{n}}

. Requiring this to be at least

0.99

means

\frac{n + 1}{2 ^{n}} \leq 0.01

, i.e.

2^{n} \geq 100 (n + 1)

. At

n = 10

2^{10} = 1024 < 1100

, which fails; at

n = 11

2^{11} = 2048 > 1200

, which works. The minimum number of bombs is

11

JEE Main · 2020 · 05 Sep 2020 · Shift 2 · Integer answer

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JEE Main · 2020 · 05 Sep 2020 · Shift 2 · Probability

Solution

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