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21.10.2014 Brain Teasers
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How many words are there in the daily edition of The Times? How many songs are stored on iPods in the UK? How many spikesare there on the back of a hedgehog? These and other provocative questions await you, in preparation for your interview withOliver Wyman.
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HEDGEHOGS, IPODS, BIRTHDAYS:Puzzles to ponder ahead of your interview.
BIRTHDAYSWhat's the probability of at least 2 players on a football field of 22 sharing the same birthday?
This is a classic problem in probability and statistics, often called the Birthday Problem. It is usually simplified by assuming 1) nobody was born onFebruary 29 and 2) people's birthdays are equally distributed over the other 365 days of the year.
The easiest way to approach this problem is to calculate the probability of the complementary event, i.e. when none of the players have the samebirthday. This probability can be calculated or approximated in several ways and one of these methods is outlined below.
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21.10.2014 Brain Teasers
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Start with showing that you understand the problem by breaking it down in smaller components, for example: The second player can't have the samebirthday as the first, i.e. there are 364 out of 365 days that are OK. The third player can't have the same birthday as player 1 or 2, so then there are363 days that are OK, and so on... Finally, all these probabilities need to be multiplied with each other to get the combined probability. Formally, itlooks like this:
Obviously, you are not expected to calculate the exact answer in an interview situation, but showing that you understand how to approach and solvethe problem are the key qualities the interviewer will look for. Once you have stated the formula above, you would probably want to give an estimateof the probability. One way to do this would be by drawing a graph with the probability on the vertical axis and the number of players on thehorizontal axis. By definition, you know that if there is only 1 player, the probability is 0 and if there are 365 players, the probability would be 1. Thisgives you the starting and ending point of the graph. To come up with the shape of the graph, you can think in terms of: What is the differencebetween having 364 and 365 players? Probably very small. However, the difference between 2 and 3 players is large (the probability almost triplessince there are now 3 combinations instead of 1). Hence, the marginal effect of adding an extra player will decrease as the number of playersincrease. The graph should look something like the one below and would allow you to estimate to probability to just below 50%.
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