Simple probability that I can't get out

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SUMMARY

The discussion centers on solving probability problems involving events A and B, birthday paradox calculations, and combinatorial choices. Key solutions include the correct set notation for events, specifically that "at most one of A or B occurs" is represented as E-(A∪B), and "exactly one of A or B occurs" as (A∪B) - (A∩B). The birthday problem is clarified with the formula P(at least two having the same birthday) = 1 - (365Pn / 365^n), where 365Pn represents permutations of unique birthdays among n people. The participants also discuss the independence of coin tosses and the geometric series involved in calculating probabilities of first heads appearing on odd tosses.

PREREQUISITES
  • Understanding of set notation in probability, specifically union (∪) and intersection (∩).
  • Familiarity with the birthday paradox and its mathematical implications.
  • Knowledge of combinatorial mathematics, particularly permutations and factorials.
  • Basic concepts of geometric series and their application in probability.
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  • Study advanced probability concepts, focusing on conditional probability and independence.
  • Learn about combinatorial proofs and their applications in solving probability problems.
  • Explore the implications of the birthday paradox in real-world scenarios, such as cryptography.
  • Investigate geometric series in-depth, particularly their convergence and applications in probability theory.
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Students of mathematics, particularly those studying probability theory, educators teaching probability concepts, and anyone interested in combinatorial mathematics and its applications in real-world problems.

rock.freak667
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Well it's been about 4 years since I've done these types of questions, so I need some help in checking if I am overthinking it or if I am just straight wrong.

I did simplify the wording a bit.

Homework Statement


a)Given 2 events, A & B, write in set notation
1)the event that at most A or B occurs
2) The event that exactly A or B occurs

The Attempt at a Solution



n=intersect, u=union

1) P(AuB)=P(A)+P(B)-P(AuB)

2) Not sure


Homework Statement


b)There are n people in a room, what is the probability that at least 2 have the same birthday?

The Attempt at a Solution



P(no 2 having the same birthday)+P(at least 2 having the same birthday)=1

P(no 2 having the same birthday)=(365/365)*(364/365)

P(no 3 having the same birthday)=(365/365)*(364/365)*(363/365)

P(no 4 having the same birthday)=(365/365)*(364/365)*(363/365)*(362/365)
.
.
.
P(no 'n' having the same birthday)=(365/365)*(364/365)*(363/365)*(362/365)*(361/365)*...*(365-n+1)/365)

It looks like the numerator should be 365! and the denominator, 365n, but I am not sure what happens to the 'n' in the numerator.


Homework Statement


c)There's an offer of a choice of 4 designs, 3 different heating systems, a garage or carport, and a patio or screened porch. How many different plans aer available?


The Attempt at a Solution



I think it would be 4*3*2*2 = 48

Homework Statement


d)A fair coin is tossed until a head appears for the first time. The tosses are independent.
Give the sameple space and what is the probability that the first head appears when the toss number is odd.


The Attempt at a Solution



S={H,TH,TTH,TTTH,TTTTH,TTTTTH,...}

Is that how it woud look?

For the odd numbered toss, it would be

P(H),P(TTH),P(TTTTH),...
= 1/2,1/8,1/32,1/128,...

which forms a GP with first term a=1/2 and common ratio r =1/4. So the sum to infinity is a/1-r

=(1/2)(1-0.25)=(1/2)*(4/3)=2/3
 
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I think you simplified the wording a bit too much on the first one. I'm not sure I understand what e.g. "at most A or B occurs". Do you mean at most one of A or B occurs? That would mean the only events that don't count is where A and B both occur. For the birthday problem, what's the probability that NONE of the n people have the same birthday? c) and d) look ok to me.
 
Dick said:
I think you simplified the wording a bit too much on the first one. I'm not sure I understand what e.g. "at most A or B occurs". Do you mean at most one of A or B occurs? That would mean the only events that don't count is where A and B both occur. For the birthday problem, what's the probability that NONE of the n people have the same birthday? c) and d) look ok to me.

sorry I left out the word one. It should bethe event that at most one of A or B occurs
The event that exactly one of A o B occursAlso if one person has a birthday, then the probability that the rest do not have that birthday is 364/365.
 
rock.freak667 said:
sorry I left out the word one. It should bethe event that at most one of A or B occurs
The event that exactly one of A o B occursAlso if one person has a birthday, then the probability that the rest do not have that birthday is 364/365.

Ok, then your whole event space, call it E, can be divided into four nonoverlapping regions, E-(AuB), A-B, B-A and AnB, right? Which combination is the answer to each part?
 
rock.freak667 said:
Also if one person has a birthday, then the probability that the rest do not have that birthday is 364/365.

Not exactly. If they all have DIFFERENT birthdays, then the first one can choose any day, the second has 364 days to choose from, the third 363, etc etc.
 
Dick said:
Ok, then your whole event space, call it E, can be divided into four nonoverlapping regions, E-(AuB), A-B, B-A and AnB, right? Which combination is the answer to each part?

I think at most one of A or B would be E(AuB), and exactly one of A or B would by E(B) or E(A) ?

Sorry I am a bit confused with it.


Dick said:
Not exactly. If they all have DIFFERENT birthdays, then the first one can choose any day, the second has 364 days to choose from, the third 363, etc etc.

Yeah, that is what I was trying with my attempt, I just didn't subtract it from 1.

1-(365!/365n)
 
rock.freak667 said:
I think at most one of A or B would be E(AuB), and exactly one of A or B would by E(B) or E(A) ?

Sorry I am a bit confused with it.

The are asking for sets, not probabilities. I am thinking of E as the set of ALL events, so A and B are subsets of E. I don't know E(A) would mean. If you mean E-(AuB) for the 'at most' part, then sounds ok. Wouldn't 'exactly one' be (AuB)-(AnB)?
 
rock.freak667 said:
Yeah, that is what I was trying with my attempt, I just didn't subtract it from 1.

1-(365!/365n)

If that's what you mean, then 1-365!/365^n is a pretty sloppy way to write it. It doesn't mean what you are describing.
 
Dick said:
The are asking for sets, not probabilities. I am thinking of E as the set of ALL events, so A and B are subsets of E. I don't know E(A) would mean. If you mean E-(AuB) for the 'at most' part, then sounds ok. Wouldn't 'exactly one' be (AuB)-(AnB)?

Then would I not have to subtract the complement of of AuB? Unless the entire event is just A and B, which I don't think I can assume.


Dick said:
If that's what you mean, then 1-365!/365^n is a pretty sloppy way to write it. It doesn't mean what you are describing.


so if (365n-365!)365n is incorrect, how do I go about doing it correctly?
 
  • #10
rock.freak667 said:
Then would I not have to subtract the complement of of AuB? Unless the entire event is just A and B, which I don't think I can assume.





so if (365n-365!)365n is incorrect, how do I go about doing it correctly?

Sure, you can subtract complement of AuB. What makes you think that would change anything? On the second one, take n=1. That's 1-365!/365. 365! is MUCH larger then 365. So that expression is negative. 1-365!/365^n is not a correct expression for your answer.
 
  • #11
Dick said:
Sure, you can subtract complement of AuB. What makes you think that would change anything? On the second one, take n=1. That's 1-365!/365. 365! is MUCH larger then 365. So that expression is negative. 1-365!/365^n is not a correct expression for your answer.

Well I am seeing that I can write the equations in my first post as factorials.


P(no 2 having the same birthday)=(365/365)*(364/365)=365!/363!3652

P(no 3 having the same birthday)=(365/365)*(364/365)*(363/365)=365!/362!3653

which can be extended to 'n' such that

P(no 'n' having the same birthday)=365!/(365-n)!365n= 365Pn/365n

So P(at least two having the same birthday)=1- 365Pn/365n

I think this should always give a number less than 1.
 
  • #12
rock.freak667 said:
Well I am seeing that I can write the equations in my first post as factorials.


P(no 2 having the same birthday)=(365/365)*(364/365)=365!/363!3652

P(no 3 having the same birthday)=(365/365)*(364/365)*(363/365)=365!/362!3653

which can be extended to 'n' such that

P(no 'n' having the same birthday)=365!/(365-n)!365n= 365Pn/365n

So P(at least two having the same birthday)=1- 365Pn/365n

I think this should always give a number less than 1.

That looks better.
 
  • #13
Dick said:
That looks better.

so it's correct then? thanks!
 

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