- A cadet fires shots at a target at distances ranging from 25 m to 90 m . The probability of hitting the target with a single shot is \(p\). When firing from a distance \(d \mathrm {~m} , p = \frac { 3 } { 200 } ( 90 - d )\). Each shot is fired independently.
The cadet fires 10 shots from a distance of 40 m .
- Find the probability that exactly 6 shots hit the target.
- Find the probability that at least 8 shots hit the target.
The cadet fires 20 shots from a distance of \(x \mathrm {~m}\).
- Find, to the nearest integer, the value of \(x\) if the cadet has an \(80 \%\) chance of hitting the target at least once.
The cadet fires 100 shots from 25 m .
- Using a suitable approximation, estimate the probability that at least 95 of these shots hit the target.