Question - Free Nights During the month of August, one guest at the Golden Nugget in Las Vegas will be selected at random to participate in a contest to win free lodging. A fair quarter will be tossed until the first head is recorded. If the first head occurs on toss x, the contestant will win x free nights' stay at the Golden Nugget. So, if a head is obtained on the first coin toss, the contest is over, and the guest wins one free night. If the first head appears on the fourteenth toss (13 tails and then a head), the guest wins 14 free nights. Theoretically, a guest could win any number of free nights, 1, 2, 3, 4, . . . , although it seems unlikely someone could win, for example, 100 free nights.
a. Use technology to model this contest. Try your simulation 10 times and record the number of free nights awarded each time. Did anyone win five or more free nights' stay?
b. Consider the event A = the guest wins five or more free nights at the Golden Nugget. Simulate the contest n = 50 times and compute the relative frequency of occurrence of the event A. Repeat this process for n = 100, 150, 200, . . . , 2000.
c. Construct a plot of the relative frequency versus the number of simulations. Describe any patterns.
d. Use your results in (b) and (c) to estimate the probability of winning five or more free nights at the Golden Nugget.
e. Find the exact probability of winning five or more free nights at the Golden Nugget.