# Complementary Binomial Distribution

Just before half term, I set my A2 students an S2 paper to complete. They’re an able group, confident in the majority of the material, but there is (part of) one topic that many of them find tricky: trying to use binomial tables when the given distribution has p>0.5. This situation may well be unique to Edexcel, but their tables only go up to 0.5.

Here’s the question that caused all the problems:

Edexcel S2, June 2012 Q8

Fortunately, I’ve drilled the students well in defining clear random variables.

• Let X = the number of customers out of 10 who ask for water. Then $X\sim B(10,0.6)$.
• Let Y = the number of customers out of 10 who don’t ask for water. Then $Y\sim B(10,0.4)$.

Then all they need to take care of is switching the ‘perspective’ on the probability questions:

• $P(X=6) = P(Y=4)$
• $P(X<9)=P(X\leq 8)=P(Y\geq 2)=1-P(Y\leq 1)$

Part (b) of this question is pretty challenging and they have to think very carefully to make sure they get every detail correct. We worked through it together:

I need to find/create a few more questions of this type. As I said to the class, it’s not enough for them to just understand it when I do it – they need to be confident in doing every step correctly for themselves.