# Negative Binomial Question

One fifth of all bars of a certain brand of chocolate contain a free ticket to go to a concert. The chocolate bars cost 60p each and the tickets cost £5 each. Six friends wish to go to the concert. They buy bars of the chocolate until they have six tickets. Let X be the number of bars they buy. (a) State the distribution of X. (b) Find the probability that the friends will buy exactly 29 bars. (c) Find the difference between the amount the friends expect to pay using their strategy and the amount they would pay if they just bought the tickets. (d) Find the probability that the friends will pay more using this strategy than if they had just bought the tickets.

I have done parts a and c correctly but I can’t get the right answer for b or d.

For b I have done X~NB(6,0.2) P(X=29)=(28C5)(0.2)^6(0.8)^17 = 0.1416 (4sf) but the correct answer is 0.0371

And I’m not sure where to start d as I thought that it would be P(X>30) but I can’t do cumulative probability with the negative binomial distribution.
For b) you want 29 with 6 wins, so dont understand your ^17?
Original post by mqb2766
For b) you want 29 with 6 wins, so dont understand your ^17?

Ah thanks, I must have typed the wrong thing into the calculator, I have the correct answer for part b now. Do you know how would I go about starting part d?
Original post by Cyanforest
Ah thanks, I must have typed the wrong thing into the calculator, I have the correct answer for part b now. Do you know how would I go about starting part d?

Dont understand why you say d) is > 30?
Original post by mqb2766
Dont understand why you say d) is > 30?

Because the concert tickets would come to £30 in total if they bought them normally as 6x£5=30 and I need to find the probability that they pay more from buying the chocolate bars than buying the tickets by themselves.
Original post by Cyanforest
Because the concert tickets would come to £30 in total if they bought them normally as 6x£5=30 and I need to find the probability that they pay more from buying the chocolate bars than buying the tickets by themselves.

Sure, but they cost 60p each.
Original post by mqb2766
Sure, but they cost 60p each.

Would I need to find >50 as 30x0.6=50? If so I’m confused as I don’t know how to find a cumulative probability using the negative binomial distribution?
Original post by Cyanforest
Would I need to find >50 as 30x0.6=50? If so I’m confused as I don’t know how to find a cumulative probability using the negative binomial distribution?

Agree that its >50. I dont know what youve covered / your level, but you can relate the negative binomial cdf to the usual binomial cdf
https://en.wikipedia.org/wiki/Negative_binomial_distribution#Cumulative_distribution_function
which is a calculator job, but dont know if thats what youre expected to do.
Original post by mqb2766
Agree that its >50. I dont know what youve covered / your level, but you can relate the negative binomial cdf to the usual binomial cdf
https://en.wikipedia.org/wiki/Negative_binomial_distribution#Cumulative_distribution_function
which is a calculator job, but dont know if thats what youre expected to do.

I’m doing Alevel further maths, I’m not sure how to relate the negative binomial distribution to the binomial distribution. Is there any further pointer you are able to give? Thanks for your help so far
Original post by Cyanforest
I’m doing Alevel further maths, I’m not sure how to relate the negative binomial distribution to the binomial distribution. Is there any further pointer you are able to give? Thanks for your help so far

What does it say in your textbook about the cumulative for the negative binomial? Im not particularly aware of this part of the syllabus and when I did a quick google for the question, part d) wasnt included. But the previous link described how the cumulative negative geometric binomial was mapped usual cumulative binomial F_binomial(k,n,p)
https://en.wikipedia.org/wiki/Binomial_distribution#Cumulative_distribution_function
and theres an example
https://en.wikipedia.org/wiki/Negative_binomial_distribution#Selling_candy
where the last part is similar to this question part, so you can work through / check the parameters here.