Quandaries and Queries


Name: Lesley
Who is asking: Parent

my daughter is having difficulty with the following formula
P(X=x) = ( n over x) px (1-p) n-x
The teacher has given them the formula but not taught them how to apply it or understand it. When asked the teacher responds that he 'only teaches at the highest level'!!

Please help.



Well, the teacher is not 'teaching' if that is his response.

It looks like he's doing what we call binomial probabilities. Suppose that you had a coin that was imbalanced so that it came up heads H with probability p (usually we have fair coins and p = 1/2) and therefore comes up tails T with probability (1-p). So for example, if the probability of an H is 1/3 then the probability of a T is 2/3. Now suppose you toss this coin n times and observe the results. Let X be the number of H's. The P(X=x) that the teacher is calculating is the probability that we get x out of n H's, and thus (n-x) T's. When you write out the sequence of H's and T's that show up in n tosses what we have is an n-letter "word" whose only letters are H and T - it might look like HHTHTTTHTHH.... HHTHTHHTT. The number of different "words" or sequences like this is what you have written as (n over x), sometimes written C(n,x). It counts the number of ways to choose x things from n things. In the probability question it is needed to choose the x spots for the H's from the n spots in our "word". The other spots are filled in with T's. We can calculate (n over x) as n!/(x!(n-x)!) where the ! is the factorial sign and it means for example that 6! = 6(5)(4)(3)(2)(1) = 720. The teacher should have explained why (n over x) is calculated this way. Now, each H appears with probability p and there are x of them; thus, since one toss of the coin doesn't affect the next toss we have the probability for the x H's as px (p to the x power) and the proability of the (n-x) T's is (1-p)(n-x). Putting this all together we have the probability of x H's in n tosses is P(X=x) = (n over x) px (1-p)(n-x).

This is a very brief explanation and should be done very slowly in class. It is absolutely pointless to produce formulas without a valid explanation - one would never know when to apply them.

Let's do one example: Suppose the probability of your favourite team winning a game is 1/4 and of losing is 3/4 (ignore ties). What is the probability that they win 3 of 5 games this season? If X counts the number of wins we want

P(X=3) = (5 over 3)(1/4)3(3/4)2
= (5!/(3!2!)(1/64)(9/16)
= 90/1024

or roughly 1/11 of the time.

Hope this helps,
Penny Nom
These problems are non-trivial and need lots of practice.


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