The letters a b c d e f g are put in a random order Define t

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

A: The letter b falls in the middle (with three before it and three after it).

B: The letter c comes after the letter b, although not necessarily immediately after it. For example, \"agbdcef\" would be an outcome in this event.

C: The letters \"def\" occur together in that order (i.e. \"gdefbca\").

1. What is p(A)?

2. What is p(B)? (Hint: first select the location for the b and c, then place the rest of the letters)

3. What is p(C)?

4. What is p(A|C)?

5. What is p(B|C)?

6. What is p(A|B)?

1. After b has been placed in the middle location there are 6 letters left to be arranged in the remaining 5 places.

therefore P(A) = 6!/ 7! = 1/7

2. letter c comes after letter b we can write this as, 6/7 + 5/7+ 4/7 +3/7 + 2/7 + 1/7 = 21/7 = 3

and remaining 5 letterrs can be places by 5! ways.

therefore P (B) = (3*5!)/7! = 1/14

3. letters def occurs together by 5 ways.

and remaining 4 letters can be arrabged by 4! ways

therfore P(C) = (5*4!)/7! = 1/42

4.P(A/C) = 2/5 since out of 5 outcomes of def occurs together only 2 ways are possible when letter b is in middle

5. P(B/C) = 3/5 since out of 5 outcomes of def occurs together only 3 ways are possible when letter c occurs after letter b.

6. P(A/B) = 1/3 since out of 3 outcomes of c occures after letter b there is only 1 possible way of b is in middle.