Perms and Combs Confusion (1 Viewer)

[mreditor16]

For some reason, I'm struggling to wrap my head around this question. Would someone be able to help explain the provided solutions and/or provide their own solution which is easier to understand? Thanks in advance, all!

Questions:

The history of an individual’s claiming record is had he had 3 quarters with 2 claims, 2 quarters with 1 claim and 15 quarters without a claim.

a) What is the probability that the insured had first 15 quarters without a claim and then 5 quarters with at least one claim?

b) What is the probability that the insured had first 15 quarters without a claim and then 2 quarters with one claim and then 3 quarters with two claims?

Solutions:

a) Use combinations, n = 20, r = 5, number of ways choosing objects: 20C5 = 15504 ; thus, probability is 1 / 15504

b) Use multinomial, n = 20, r1 = 15, r2 = 3, r3 = 3, number of ways choosing objects = 20! / (15! 2! 3!) = 155040 ; thus, probability is 1 / 155040

 

[InteGrand]

For some reason, I'm struggling to wrap my head around this question. Would someone be able to help explain the provided solutions and/or provide their own solution which is easier to understand? Thanks in advance, all!

Questions:

The history of an individual’s claiming record is had he had 3 quarters with 2 claims, 2 quarters with 1 claim and 15 quarters without a claim.

a) What is the probability that the insured had first 15 quarters without a claim and then 5 quarters with at least one claim?

b) What is the probability that the insured had first 15 quarters without a claim and then 2 quarters with one claim and then 3 quarters with two claims?

Solutions:

a) Use combinations, n = 20, r = 5, number of ways choosing objects: 20C5 = 15504 ; thus, probability is 1 / 15504

b) Use multinomial, n = 20, r1 = 15, r2 = 3, r3 = 3, number of ways choosing objects = 20! / (15! 2! 3!) = 155040 ; thus, probability is 1 / 155040

This question is equivalent to thinking about it in terms of randomly arranging 15 white balls (quarters with no claims), 2 blue balls (quarters with one claim), and 3 black ball (quarters with 3 claims) in a straight line.

The first Q. is asking us what's the probability that when we arrange them, the first 15 balls are all white. This can be done in (15!•5!) ways (arrange first 15 balls as white balls in any order, then arrange the last 5 in any order) so the answer is (15!•5!)/(20!), which is what the answers had.

The second is asking for the probability that the first 15 balls are white, then next two are blue, then last three are black. This can be done in (15!•2!•3!) ways (similar reasoning to first Q.) Hence the answer is (15!•2!•3!)/(20!), which is what the answers had.

 

[mreditor16]

This question is equivalent to thinking about it in terms of randomly arranging 15 white balls (quarters with no claims), 2 blue balls (quarters with one claim), and 3 black ball (quarters with 3 claims) in a straight line.

The first Q. is asking us what's the probability that when we arrange them, the first 15 balls are all white. This can be done in (15!•5!) ways (arrange first 15 balls as white balls in any order, then arrange the last 5 in any order) so the answer is (15!•5!)/(20!), which is what the answers had.

The second is asking for the probability that the first 15 balls are white, then next two are blue, then last three are black. This can be done in (15!•2!•3!) ways (similar reasoning to first Q.) Hence the answer is (15!•2!•3!)/(20!), which is what the answers had.

Thanks for that, Integrand. Could you shed any light on the reasoning/approach used in the provided solutions?

 

[InteGrand]

Thanks for that, Integrand. Could you shed any light on the reasoning/approach used in the provided solutions?

Welcome! For the solutions, this is basically their reasoning (I'll just use the ball analogy again).

a) We want the probability that the last five balls to be non-white. Since there are 20 balls to choose from to place in the last 5, we can choose which ones go in the last five in 20C5 ways. Exactly 1 of these choices corresponds to the last five balls all being non-white (namely the choice where we picked all 5 of the non-white balls in our set of 20). Hence the probability we placed no whites in the last five is 1/(20C5), which is thus the answer.

b) We want the probability that the balls will form the following pattern: 15W, 2Blue, 3Black. Note that the total number of possible configurations is (20!)/(15!•2!•3!) (treating balls of the same colour as identical, typical HSC perms and combs calculation). Exactly one configuration is the one we want (namely 15W, 2Blue, 3Black), so the desired probability is (number of desired configurations)/(total number of configurations) = (1)/((20!)/(15!•2!•3!)) = solutions' answer.

(Note for part b), "configurations'' just meant patterns of colours, so balls of same colour were essentially identical. The reason that doing the probability this way yields the same answer compared to calculating the probability by considering the balls as non-identical is that the total number of ways of achieving any arrangement of colours in the second method is a constant multiple k of that in the first method for the same order of colours, regardless of what the order is. In fact, this k is 15!•2!•3!. When we calculate the probability, since we are essentially dividing two numbers representing no. of arrangements, the k's would be present if we used the second method, but they'd cancel out when dividing, thus yielding the same probability.)

 

[eyeseeyou]

I'm just wondering but what is it about perms and combs which people hate? Personally I've never learnt it so I wouldn't know

 

[jathu123]

I'm just wondering but what is it about perms and combs which people hate? Personally I've never learnt it so I wouldn't know

As with me, I find the wording of questions difficult to understand, maybe that's cause I haven't practice enough questions haha.

 

[eyeseeyou]

As with me, I find the wording of questions difficult to understand, maybe that's cause I haven't practice enough questions haha.

Lol fair enough. I can't even comprehend the questions myself

 

[braintic]

I'm just wondering but what is it about perms and combs which people hate? Personally I've never learnt it so I wouldn't know

When a topic is not fully understood, people survive by recognising different types of common questions.
People struggle with this topic because they struggle to link new questions to questions they have seen before.