Probability - the Australian Mathematical Sciences Institute

{32} • ProbabilityExercise 11abcdA boat offers tours to see dolphins in a partially enclosed bay. The probability ofseeing dolphins on a trip is 0.7. Assuming independence between trips with regardto the sighting of dolphins, what is the probability of not seeing dolphins:i on two successive trips?ii on sevens trips in succession?A machine has four components which fail independently, with probabilities of failure0.1, 0.01, 0.01 and 0.005. Calculate the probability of the machine failing if:i all components have to fail for the machine to failii any single component failing leads to the machine failing.Opening the building of an organisation in the morning of a working day is a responsibilityshared between six people, each of whom has a key. The chances that theyarrive at the building before the required time are, respectively, 0.95, 0.90, 0.80, 0.75,0.50 and 0.10. Do you think it is reasonable to assume that their arrival times are mutuallyindependent? Assuming they are, find the chance that the building is openedon time.In the assessment of the safety of nuclear reactors, calculations such as the followinghave been made.In any year, for one reactor, the chance of a large loss-of-coolant accidentis estimated to be 3 × 10 −4 . The probability of the failure of the requiredsafety functions is 2×10 −3 . Therefore the chance of reactor meltdown viathis mode is 6 × 10 −7 .What do you think of this argument?The next example illustrates the somewhat strange phenomenon of events that are notmutually independent but nevertheless satisfy equation (∗).ExampleConsider the random procedure of tossing a fair coin three times. Define the events:• A = “at least two heads”• B = “the last two tosses give the same result”• C = “the first two results are heads or the last two results are tails”.