SECTION 3.6

1.

In a study, the various stages of sleep for a large group of people were monitored in 30-second intervals, or "epochs.\" For each epoch, sleep stage was categorized as Wake, REM, or Non-REM. The accompanying table provides a summary of the results. Each cell of the table gives the number of epochs that occurred when transitioning from the previous sleep stage to the current sleep stage. Complete parts a through e below. A Click the icon to view the sleep stage data table. a. Given that the previous sleep stage for the epoch was the Wake state, what is the probability that the current sleep stage is REM? The probability is | | (Type an integer or a decimal. Round to three decimal places as needed.) Sleep Stage Data Table Previous Sleep Stage Current Stage Non-REM REM Wake & Non-REM 31,773 194 1,866 REM 369 7,635 170 Wake 1,668 306 5,281 A v Totals 33,810 8,135 7,317 The most likely scenario for an accident for a natural gas pipeline is natural-gas leakage from a hole in the pipeline. The probability that the leak ignites immediately (1) causing a jet fire is 0.17. If the leak does not immediately ignite, it may result in the delayed ignition (D) of a gas cloud. If there is no delayed ignition, the gas cloud will disperse harmlessly (H). Given no immediate ignition, the probability of delayed ignition causing a flash fire is 0.17. Suppose a leak occurs in the natural-gas pipeline. The probability that either a jet fire or a flash fire will occur is D (Round to four decimal places as needed.) The outcomes of two variables are (Low, Medium, High) and (On, Off), respectively. An experiment is conducted in which the outcomes of each of the two variables are observed. The probabilities associated with each of the six possible outcome pairs are given in the accompanying two-way table. Consider the events below. Complete parts a through h. A: {On} B: {Medium or On} C: {Off and Low} D: {High} a. Find P(A). P(A)= D (Type an exact answer in simplified form.) Low Medium High On Off 052 0.25 011 0.05 0.04 0.03 Zoologists investigated the likelihood of fallow deer bucks fighting during the mating season. During a 270-hour observation period, the researchers recorded 205 encounters between two bucks. Of these, 164 involved one buck clearly initiating the encounter with the other. In these 164 initiated encounters, the zoologists kept track of whether or not a physical contact fight occurred and whether the initiator ultimately won or lost the encounter. (The buck that is driven away by the other is considered the loser.) Suppose we select one of these 164 encounters and note the outcome (fight status and winner). FH Click the icon to view a summary of the 164 initiated encounters. a. What is the probability that a fight occurs and the initiator wins? The probability is| |. (Round to the nearest thousandth as needed.) Data table Initiator No Clear Initiator Wins Winner Loses Fight 24 24 15 63 No Fight 80 11 10 101 Totals 104 35 25 164 Totals