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Two nations, Greater Brobdingnag* and Lesser Brobdingnag (or G and L, for short) are at war.  

G has more sophisticated weaponry, which forces L to rely on guerilla tactics.  When they fight a battle, the odds are 2 to 1 that G will win. (i.e. on average, out of 3 battles, G will win twice, while L will only win once.)

Suppose that G and L fight 2 battles.  What is the probability that G will win at least one battle?

Here are the steps to accomplish this:

1. Let (for example) the ordered pair of letters GL denote the event that G wins the first battle, and that L wins the second battle

   GG, of course, denotes the event that G wins both battles, and so on.

2. The sample space for the experiment consists of a number of sample points, E_i, each of which is one possible outcome of the two battles.  Write down all the possible sample points:

   E₁: GG,  E₂: GL, ... etc.

3. Determine the probability that G will win a single battle, and the probability that L will win a single battle. (I recommend you use simple fractions here.)

   Based on these probabilities, determine the individual probability of each of the events E_i occurring.  

   If you add up the probabilities for all the events E_i, what value do you come up with?

4. Denote as C the event of interest (that G wins at least one battle out of two fought).

   Which of the E_i satisfies the conditions for event C?

   From this information, and the respective probability of each event, what is the probability, P(C), of event C occurring?

* Strange names for countries don't you think?  Do you have any idea where the name Brobdingnag came from?