Binomial Probability (1 Viewer)

[gr_111]

Question from SGHS 2012 Trial:

12b) The probability that it rains on any particular day in London is 2/3.
ii) What is the probability that it will rain on only two days during a whole week in London, and that these two days are consecutive?

My working:
There are six ways of having two consecutive days in a week, and for each of those ways the probability is given by

which means my final answer is

However, the answers do not have the "7C2" included, have I stuffed up or are the answers wrong?

 

[Aysce]

I think you're correct because since 2 days are raining in one week, it has to be 7C2 (2/3)^2 (1/3)^5

I don't see anything wrong with it though.

 

[Carrotsticks]

Think about what the 7C2 is there for in the first place.

Or more generally, think about WHY we have nCk when we do Binomial Probability problems.

 

[gr_111]

Well, the nCk is there because it tells us the number of ways we can achieve the desired of outcomes of k favourable results and n-k unfavourable results.

So are you saying i should or shouldn't have it there?

 

[Aesytic]

the nCk should not be there; what that's there for normally is to account for the varying ways in which you can have 2 days raining and 5 days not raining out pf 7 days
the (2/3)^2*(1/3)^5 by itself (without the 7C2) shows the probability of having 2 days raining and 5 days not raining for only one particular arrangment. now we know that for it to rain on 2 consecutive days, then there are only 6 different ways of having this, so it's 6*(2/3)^2*(1/3)^5

you don't have 7C2*(2/3)^2*(1/3)^5 is the probability of it raining on any 2 days out of 7, but it doesn't take into account the restriction that we have (raining on 2 consecutive days)

 

[Carrotsticks]

Well, the nCk is there because it tells us the number of ways we can achieve the desired of outcomes of k favourable results and n-k unfavourable results.

So are you saying i should or shouldn't have it there?

Exactly correct.

So by having the 7C2 there, you are counting ALL ways of having 2 rainy days in a week, not just 2 consecutive days.

 

[gr_111]

ahah!!!! yep i see it now thanks guys!