Probability Question - Need help

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The probability question involves three boxes with different combinations of chocolate and vanilla cupcakes. After selecting a chocolate cupcake, the focus shifts to determining the probability that the remaining cupcake in the same box is also chocolate. The calculation uses conditional probability, leading to the conclusion that the probability of the second cupcake being chocolate is 2/3. This is derived from the ratio of the probability of selecting the chocolate box given a chocolate cupcake has been picked to the overall probability of picking a chocolate cupcake. The discussion emphasizes the importance of understanding conditional probability in solving such problems.
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Probability Question - Need help please!

There are three boxes with 2 cupcakes in each one of them:

First box: 2 chocolate cupcakes
Second box: 1 chocolate, 1 vanilla
Third box: 2 vanilla cupcakes

A person randomly opens up a box and selects one cupcake. It is a chocolate cupcake. She then pulls out the remaining cupcake in the same box. What is the probability that it is a chocolate cupcake?

Ok here is what I think...
Since the question is only asking for the probability that the second cupcake in the box is chocolate, we don't have to worry about the first step of picking the right box. Therefore, the probability should be 1/2 because you can either pick a vanilla or a chocolate, but it doesn't really sound convincing...
 
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firstwave said:
There are three boxes with 2 cupcakes in each one of them:

First box: 2 chocolate cupcakes
Second box: 1 chocolate, 1 vanilla
Third box: 2 vanilla cupcakes

A person randomly opens up a box and selects one cupcake. It is a chocolate cupcake. She then pulls out the remaining cupcake in the same box. What is the probability that it is a chocolate cupcake?
It's a conditional prob. question, "what is the prob. of the box being the choc. box given a choc cake has been picked?"

P(Choc box|a choc cake has been picked) = P(Choc box and a choc cake has been picked)/P(a choc cake has been picked) = P(Choc box)/(3/6) = (1/3) / (1/2) = 2/3.

See this example.
 
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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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