Question

In a popular tale of wizards and witches, a group of them finds themselves in a room with doors which change position, making it impossible to determine which door is which when the room is entered or reentered. Suppose that there are 4 doors in the room. One door leads out of the building after 3.5 hours of travel. The second and third doors return to the room after 5 and 3.5 hours of travel, respectively. The fourth door leads to a dead end, the end of which is a 1.5 hour trip from the door.

If the probabilities with which the group selects the four doors are 0.2, 0.1, 0.2, and 0.5, respectively, what is the expected number of hours before the group exits the building?

Answer 1

The expected number of hours before the group exits the building is calculated by finding the expected value for each door based on the given probabilities and travel times, resulting in a total of 2.65 hours.

Step-by-step explanation:

To calculate the expected number of hours before the group exits the building, we multiply the number of hours associated with each door by the probabilities of choosing that door and then we sum the results. This calculation is known as finding the expected value in probability and statistics.

The expected time (E) for each door is as follows:

Door 1 (leads out): E1 = Probability (P1) * Time (T1) = 0.2 * 3.5 hours

Door 2 (returns): E2 = P2 * T2 = 0.1 * 5 hours

Door 3 (returns): E3 = P3 * T3 = 0.2 * 3.5 hours

Door 4 (dead end): E4 = P4 * T4 = 0.5 * 1.5 hours

Now we sum these expected times for all doors to get the total expected time (Etotal):

Etotal = E1 + E2 + E3 + E4

= (0.2 * 3.5) + (0.1 * 5) + (0.2 * 3.5) + (0.5 * 1.5)

= 0.7 + 0.5 + 0.7 + 0.75

= 2.65 hours

The expected number of hours before the group exits the building is 2.65 hours.