There is a new toll ring around the town Autopol. There are some initial hiccups, so the company running the toll booths only get to check on one in every four cars passing through the toll gates. But the cars that have not paid when checked get a major fine. The cruisers Hans Rust-Holch and Fritz Königsegg want to test the new system, and discuss the probabilities of getting caught dodging the toll payments.

(a) What is the probability that they will be caught dodging if they do it once (1 time)?

(b) What is the probability that they will be caught dodging twice if they dodge 5 times?

(c) What is the smallest number of times they must dodge the toll payment for the expected number of times to get caught to be at least 2 ? What is the standard deviation?

(d) What is the largest number of times they can dodge the payment before the probability of getting caught at least once exceeds \(50 \%\) ?

(e) After 12 dodgings, Hans and Fritz have been caught 4 times. Seven of the 12 times they dodged the payment, Hans sat in the rear, reading