## Friday, May 11, 2012

### An Explanation of the Probability Problem

First, here's a restatement of yesterday's probability question :

Compare these 2 probabilities and explain method:

(a) Prob of rolling exactly 3 sixes in 5 rolls of a fair die.

(b) Prob of rolling exactly 3 sevens in 5 rolls of a pair of fair dice

Discussion :
Both are examples of binomial probability because they involve repeated independent  trials each of which has 2 outcomes. The following explanation is intentionally detailed and 'repetitious'.

The prob of a 6 on each roll is 1/6. Each roll produces only 2 outcomes, either a 6 (prob=1/6) or not a 6 (prob = 5/6).

The prob of a 7 on each roll of a pair of dice is 6/36 or 1/6. Each roll of the pair has only 2 outcomes, either a 7 (prob=1/6) or not a 7 (prob=5/6).

Therefore, the probabilities of getting 3 successes in 5 trials is the same. Since the question asks for a comparison, we're done.

The actual prob is C(5,3)(1/6)^3•(5/6)^2 where C(5,3) is the 'MathNotation' for the number of ways of arranging 5 objects, one group of 3 identical objects and a separate group of 2 identical objects. This is not the usual way of defining combinations but I like this interpretation.

I guess the QuadAnagram was a bit challenging. Here's a hint for the ending:

...he's a bored L---R.