2 Questions: Odds and Probability

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The discussion revolves around two probability questions regarding odds and birthday coincidences. The first question asks for the combined odds of either team A or team B winning a championship, given their individual odds of 1:5 and 2:13, respectively. The second question seeks the probability that exactly two out of eight friends share the same birthday. Participants are encouraged to provide detailed explanations and show their work to assist in solving these problems. Clear calculations and methodologies are essential for accurate answers.
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Hey all,

I was going over some questions in my textbook when I came across these two questions. I could not get the answer in the back of the book (possibly because there is an error) or I did something wrong. I'd appreciate it if anybody can help me out.

1) If the odds in favor of team A winning the championship is 1:5, and the odds in favor of team B winning the championship is 2:13, what are the odds in favor of either team A or team B winning the championship.

2) What is the probability that exactly two out of eight friends will have the same birthday?


I'd appreciate any input

Thanks
 
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you have to show some work, what have you tried?
 

Thread 'Finding the number of ways to arrange identical balls in a circle (3 different colors)'

I tried to combine those 2 formulas but it didn't work. I tried using another case where there are 2 red balls and 2 blue balls only so when combining the formula I got ##\frac{(4-1)!}{2!2!}=\frac{3}{2}## which does not make sense. Is there any formula to calculate cyclic permutation of identical objects or I have to do it by listing all the possibilities? Thanks
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Thread 'Greatest possible value of a constant in polynomial'

Since ##px^9+q## is the factor, then ##x^9=\frac{-q}{p}## will be one of the roots. Let ##f(x)=27x^{18}+bx^9+70##, then: $$27\left(\frac{-q}{p}\right)^2+b\left(\frac{-q}{p}\right)+70=0$$ $$b=27 \frac{q}{p}+70 \frac{p}{q}$$ $$b=\frac{27q^2+70p^2}{pq}$$ From this expression, it looks like there is no greatest value of ##b## because increasing the value of ##p## and ##q## will also increase the value of ##b##. How to find the greatest value of ##b##? Thanks
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