October Puzzle Periodical - CyberChance

By Nicholas R., NSA, Applied Research Mathematician

Eddie, Layne, Kurt, and Chris are locked in a heated game of CyberChance which only one of them will win. Their fans are on the edge of their seats: everyone knows that in CyberChance, things can change at any moment. One fan, who prefers Kurt or Chris, is feeling worried since there is only a 4-in-10 chance that one of them will win. An Eddie fan brags that Eddie is twice as likely as Layne to come out on top. Another Eddie fan concurs, and adds the following observation: Chris is Eddie’s main competition. He figures that, given that Chris doesn’t win, Eddie has a 4-in-7 chance of winning. A Kurt fan has been quietly standing in the corner, listening to all the other fans, and wonders: What chance does Kurt have of winning CyberChance?

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Solution

Let E, L, K, and C be the chances that Eddie, Layne, Kurt, and Chris win the game, respectively. There is a 100% chance that one of them will win, so E + L + K + C = 100%. We know that K + C, the probability that Kurt or Chris will win, is 40%. This tells us that E + L = 100% - 40% = 60%. Now, we also know that the probability that Eddie will win is twice the probability that Layne will win. In numbers, this means that E = 2*L, i.e. E – 2*L = 0%. Knowing these two facts lets us find out what Eddie’s chance of winning is: add the equation E + L = 60% twice to the equation E – 2*L = 0%. This tells us that 3*E = 120%. Therefore E = 40%. We are almost there. We know that the conditional probability that Eddie wins, given that Chris loses, is 40%/70%. This probability is equal to the probability that Eddie wins and Chris loses, divided by the probability that Chris loses. But notice: CyberChance has only one winner, so the probability that Eddie wins and Chris loses is the same as the probability that Eddie wins. Also notice that the probability that Chris loses equals the probability that anybody else wins. In numbers, the probability that Chris loses is 100% - C. So we have the following equation:

40%/70% = E / (100% – C) = 40% / (100% – C)

Solving for C tells us that C = 30%. Recall that K + C = 40%. Therefore K = 10%. So Kurt has a measly 10% chance of winning CyberChance. No wonder the Kurt fan was sulking in the corner!