1.Identify the assumptions required to calculate the probability that four researchers would all be...

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1.Identify the assumptions required to calculate the probability that four researchers would all be assigned middle seats on three flights.		2.Discuss whether these assumptions are reasonable and what effect any violations to these assumptions may have on the stated probability of one in 540 million.How “random” is Ryanair's seating allocation? ASK A STATISTICIAN  How “random” is Ryanair’s seating allocation? Jennifer Rogers calculates the probability of a suspect seating arrangement on a plane Irish airline Ryanair has come under fire for its so-called “random seat allocation” policy for passengers  who choose not to pay to reserve a  seat. The issue came to a head when 15  people on a hen party (or bachelorette  party) were separated on a recent  flight, with all being allocated middle  seats (bit.ly/2v7RDdD). But what is the  probability of that happening if seats  are indeed allocated at random? Calculating the hen party probability  is impossible without knowing what  seats were available at the time of check- in, which is when the random allocation  is said to take place. But I can make an  attempt at answering the question based  on some work I did for Watchdog, a  consumer affairs show broadcast by the  BBC. The producers were interested in  the Ryanair policy as there had been a  number of complaints about it.  Watchdog sent four of its researchers  to travel together on four different  flights – and on every single one of  these flights, each researcher was  allocated a middle seat. I was given  access to the number of seats available  at check-in for each flight and, using  that information, I could calculate the  chances of all four researchers being  randomly allocated middle seats on  all four flights. For example, on their  Manchester to Dublin flight there  were 65 seats available at the time  of check-in, 15 of which were middle  seats. Therefore, we can calculate the  probability of being allocated four  middle seats on this flight as × × × = 15 14 13 12 0.002 65 64 63 62 or one in 500. Considering how many  groups of four will fly with Ryanair  every day, we perhaps should not be  surprised if one in every 500 groups is  randomly allocated all middle seats.  But things get more interesting when  you consider the fact that the four  researchers were all seated in middle  seats on three further flights. Carrying  out similar calculations as above for  the other flights, and then combining  these probabilities together, I was able  to calculate the probability of four  researchers being allocated middle  seats on their four flights, if the seating  allocation was indeed random, as  one in 540 million. A tiny probability!  Indeed, there is a much greater chance  (one in 45 million) of winning the  jackpot on the UK National Lottery. Be aware, though, that a small  probability does not always constitute  a rare event. My chance of winning  the lottery may be slim, but I’m not  surprised when someone wins the  jackpot because so many people  play each week. However, Ryanair’s  published facts and figures state that  they will carry 130 million passengers  this year, meaning that our “random”  seat allocation scenario not only has a  tiny probability, it is also a rare event  (see “How surprising?”). Ryanair claims that middle seats  are being randomly allocated so often  because people overwhelmingly  favour the aisle and window seats  when paying to reserve. But the data  collected by Watchdog showed that  on these four flights there were fewer  available middle seats than window or  aisle seats. (Also, note that I am writing  this article on a Ryanair flight to Rome,  sitting in a middle seat, and at the time  of check-in there were only 18 middle  seats available, compared with 23  window and 25 aisle seats). But let us go back to our unfortunate  hen party: if we were to assume that a  third of the seats available at check-in  were middle seats, the probability that  all 15 party-goers would be allocated  middle seats is (1/3)15, or close to one in  14 million. n Dr Jennifer Rogers is  director of statistical  consultancy services  in the Department of  Statistics, University  of Oxford. How surprising? Not all rare events are surprising. Weaver  defined the surprise index of an event E as  the ratio of the expected value of all the  probabilities to the probability of E. He also  provided the levels of surprise shown in Table  1.1 For the Ryanair hen party problem, the  underlying model is a binomial distribution  with parameters n = 15, p = 1/3, and the  probability of observing 15 middle seats in a  sample of size 15 is p 15.  Redheffer gives the expression for the  surprise index of a binomial distribution.2  For 15 middle seats in a sample of 15 party- goers, the index score is 2 203 439 – that  is, the expected value of all the model’s  probabilities is more than 2 million times larger  than the probability of observing 15 middle  seats randomly allocated. This is rare and very  surprising, though not quite a miracle. n n	 Mario Cortina Borja is chairman of  the Significance editorial board, and  professor of biostatistics at University  College London. References 1. Weaver, W. (1948) Probability, rarity, interest and  surprise. Scientific Monthly, 67, 390–392. 2. Redheffer, R. M. (1951) A note on the surprise index.  Annals of Mathematical Statistics, 22, 128–130. TABLE 1 Weaver’s interpretation of surprise values.

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How “random” is Ryanair's seating allocation? ASK A STATISTICIAN How “random” is Ryanair’s seating allocation? Jennifer Rogers calculates the probability of a suspect seating arrangement on a plane...

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