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A gambling company has these two handicap events related with today's Real Madrid versus Atlético de Madrid match:

  • Real Madrid -1: 5.00
  • Atlético de Madrid +3: 1.04

Are these events dependent or independent? It doesn't seem clear to me. If they are independent I can compute the joint probability like: 1/5 * 1/1.04 But if they are not I will need the conditional probability of one of them.

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  • $\begingroup$ Can you explain them a bit further? Does the first mean Real to win but giving a one goal start? $\endgroup$
    – mdewey
    Commented May 28, 2016 at 16:06
  • $\begingroup$ Yes. RM -1 happens if 2-0, 3-1, 4-2 and so on (RM vs ATM). $\endgroup$
    – hipoglucido
    Commented May 28, 2016 at 17:37

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From your description

  • "RM: -1" is the event that Real Madrid would win by more than one goal
  • "AM: +3" is the event that Atlético de Madrid would lose by fewer than three goals, or draw, or win

These events were not independent. In particular if the first event did not happen, you could be certain that the second would happen. Similarly if the second event did not happen, you could be certain that the first would happen

You should not expect the gambling company to offer fair odds. But if they had been fair then you could have said that there were three possible outcomes:

  • Real Madrid win by three or more goals, with probability $1-\frac1{1.04}\approx 0.03846$
  • Real Madrid wins by exactly two goals, with probability $\frac{1}{5}-1+\frac1{1.04}\approx 0.16154$
  • Real Madrid wins by exactly one goal, or there is a draw or Atlético de Madrid wins, with probability $1-\frac15 = 0.8$

In fact the match on 28 May 2016 was a draw after $90$ minutes and after extra time, with Real Madrid winning on penalties, so the third possibility occurred and the $1.04$ bet paid off, albeit not very generously

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