12.1 Calculating conditional probabilities - random permutations.

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The letters {a, b, c, d, e, f, g} are put in a random order. Each permutation is equally likely. Define the following events:

• A: The letter b falls in the middle (with three before it and three after it)
• B: The letter c appears to the right of b, although c is not necessarily immediately to the right of b. For example, "agbdcef" would be an outcome in this event.
• C: The letters "def" occur together in that order (e.g. "gdefbca")

(a)

Calculate the probability of each individual event. That is, calculate p(A), p(B), and p(C).

(b)

What is p(A|C)?

(c)

What is p(B|C)?

(d)

What is p(A|B)?

(e)

Which pairs of events among A, B, and C are independent?