Probability maths!!?

Question

On a line of bus, if one is controlled without having ticket, the fine costs 150€. The probability of being controlled a certain day is 1/10. The fact of being controlled a certain day is independent because of being another day.

which of the following is true...??? justify.................................................

(A) the probability of being controlled the every day during 5 days consecutive is 1/50 ..................................................................

(B) the probability of being controlled exactly once during 5 days consecutive is (5 × 9^4)/10^5 .........................................................

(C) a frauder who never buys ticket pays on average, with the fines which he has from time to time, 15€ per day ...................................................

(D) Over 80 days consecutive, a passenger is controlled on average 8 times. ........................

(E) Over 80 days consecutive, the probability never of not being controlled is 0.

Answer 1

(B), (C) and (D) are true, I think; I didn't think very hard about it and don't quite remember everything I studied in school.

(B) is based on P(ticketoneday)*P(noticketstheotherdays)*(5choose1)
=1/10 * (9/10)^4 * (5!/(1!4!)
=5*9^4/10^5

but for all I know it could be a trick question which is easy to do with probability, you have to be careful

c) is a no-brainer, that's the mathematical expectation over time, =P(outcome) * payoff

d) is the same thing but over less time.

e) is obviously false. (1/10)^80 is not equal to 0.

a) is also a silly one, P (being checked each day) = (1/10)^5.

Answer 2

A) False - probabilities are multiplied when calculating multiple occurances, therefore: 1/10^5 or 1/100000

B) Probability of being controlled specifically on 1st day only is 1/10*(9/10)^4. Since the one time could be ANY day we must multiply this by 5, so: 5/10(9/10)^4 = 5*9^4/10^5, so True

C) True, although the more often he rides, the closer the actual average will be to 15

D) This is a strange question: a passanger will be fined x times over 80 days. There is no average. However if n passengers (n=big number) ride 80 days, then true: on average, each will be controlled 8 times.

E) False, although it is approaching zero: (9/10)^80 = very small number

Answer 3

A. wrong. Being checked 5 days in a row, the chance is 1/10 * 1/10 ... 5 times so 1/100000

B. The probability is:
4 days not being checked one once being checked ON THE LAST DAY ONLY =
9/10 * 9/10 * 9/10 * 9/10 * 1/10
Now add that the one time you were checked can occur on any day, so multiply by 5:
( 5 * 9*9*9*9 ) / (10*10*10*10*10) = the correct answer.
TRUE.

C. expected outcome is the price of winning times the chance you win. Answer is correct.

D. Yes. The word that makes this answer true is 'average'.

E. No, the probability of not being controlled will never be 0.