3 In air traffic management, air traffic controllers send radio messages to pilots. On receiving a message, the pilot repeats it back to the controller to check that it has been understood correctly.
At a particular site, on average \(4 \%\) of messages sent by controllers are not repeated back correctly and so have been misunderstood. You should assume that instances of messages being misunderstood occur randomly and independently.
- Find the probability that exactly 2 messages are misunderstood in a sequence of 50 messages.
- Find the probability that in a sequence of messages, the 10th message is the first one which is misunderstood.
- Find the probability that in a sequence of 20 messages, there are no misunderstood messages.
- Determine the expected number of messages required for 3 of them to be misunderstood.
- Determine the probability that in a sequence of messages, the 3rd misunderstood message is the 60th message in the sequence.