1.When someone buys a ticket for an airline flight, there is a 0.0995 probability that the person will not show up for the flight (based on data from an IBM research paper by Lawrence, Hong, and Cherrier). An agent for Air America wants to book 24 persons on an airplane that can seat only 22. If 24 persons are booked, find the probability that not enough seats will be available. Is this probability low enough so that overbooking is not a real concern?
p = probability that a person will show up on a flight = 1 - 0.0995 = 0.9005, and q = 0.0995n = 24
P(x) = nCx p^x q^(n - x)
P(x > 22) = P(23) + P(24) = 24C23 0.9005^23 0.0995^1 + 24C24 0.9005^24 0.0995^0 = 0.2952
Probability that not enough seats are available = 0.2952
This probability is not low-enough not to merit concern.
2.Neuroblastoma, a rare form of malignant tumor , occurs in 11 children in a million, so its probability is 0.000011. Four cases of neuroblastoma occurred in Oak Park, Illinois, which had 12,429 children a.assuming that neuroblastoma occurs as usual, find the mean number of cases in groups of 12,429 children. b.find the probability that the number of neuroblastoma cases in group a of 12,429 children is 0 or 1. c.what is the probability of more than one case? d.does the cluster of four cases appear to be attribute to random chance? why?
(a) Mean = np = 12429 * 0.000011 = 0.1367(b) p = 0.000011, q = 1 - p = 0.999989
P(x) = nCx p^x q^(n - x)
P(0 or 1) = P(0) + P(1) = 12429C0 0.000011^0 + 0.999989^12429 + 12429C1 0.000011^1 + 0.999989^12428 = 0.9914
(c) P(x > 1) = 1 - P(x ≤ 1) = 1 - 0.9914 = 0.0086
(d) Yes, it does because probability of occurrence of more than 1 case is very low, 4 cases is due to pure chance alone.