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Two players take turns tossing a fair coin. The first player to obtain “heads” wins the game.?
Determine the probability that the player who tosses first will win the game?
QUESTION 2)
For S = {1, 3, 5, 7}, A = {3, 5, 7} ∈ A, and B = {1, 3, 5} ∈ A, where A is an algebra, list the required elements of A.
Question 1)
One quick way to solve a question such as this is the following.
Let p denote the probability that the 1st player, say A, wins. What is the probability 1-p that the 2nd player B wins?
B has a chance of winning *only* when A "misses" (i.e. gets a Tail) on the first toss. This happens exactly half the time, and when it does happen, B finds himself in the position of A at the start of the game. So from that point on, B's probability of winning is p. But this happens only half the time, since it was contingent on A missing on the first toss.
So
1 - p = (1/2) (p),
whence p = 2/3.
More formally, using conditional probability,
P(B wins) = P( (A misses on toss 1) AND B wins )
= P(A misses on toss 1) P( B wins | A misses on toss 1 ),
so
1 - p = (1/2) p etc.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Question 2)
First, suppose S = {x, y, z}, A = {y,z}, B = {x,y}.
C = A ∩ B = {y} is in our algebra G (renaming it to avoid confusion with names of sets!).
So D = {z} = A - C, E = {x} = B - C are in the algebra G also.
Since G contains all the singleton subsets of S, G must be the power set of S.
Now in our question, you will notice that the y above is just the two elements 3 and 5 considered together. So, mirroring the argument above, you will see that
G is the set of all subsets of S that either contain both 3 and 5 or neither of the two. In particular, G will have 8 elements.
One can read more about algebras over sets at the following reference link.
http://en.wikipedia.org/wiki/Algebra_over_a_set
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