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Two shops have cupcakes. Shop $A$ have $2$ blue cupcakes and $3$ chocolate cupcakes. Shop $B$ has $2$ blue cupcakes and $5$ chocolate cupcakes. Suppose we randomly choose a cupcake from $A$ and transfer it to $B$. What is the probability that the cupcake we choose is blue from $B$?

My attempt at this was to say that there are two cases:

  1. If a chocolate cupcake was transferred then we would have $8$ total cupcakes at shop $B$, so choosing a chocolate cupcake is $\frac{5}{8}$.

  2. If a blue cupcake was transferred then we still have $8$ total cupcakes at shop $B$, so choosing a blue cupcake becomes $\frac{2}{8}$.

All together we have: $\frac{3}{8}\cdot \frac{2}{7} + \frac{2}{8} \cdot \frac{5}{7} = \frac{2}{7}$.

Is this the correct way to think about this problem?

DippyDog
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    I don't understand your calculation. Where do all the $7's$ come from? $A$ starts with $5$ cupcakes, so I'd have expected to see some $5's$ but not $7's$ in the denominators. – lulu Mar 15 '21 at 16:49
  • I thought the $7$'s are from before the transfer... Should the probability the change a blue cupcake is taken from $A$? – DippyDog Mar 15 '21 at 16:51
  • The question is not clear to me: Is it that the cupcake is blue given cupcake is originally from B or something else? Have you translated the exercise? – callculus42 Mar 15 '21 at 16:53
  • @callculus, we randomly choose a cupcake from $A$ and transfer it to $B$. So the cupcake is either blue or chocolate. – DippyDog Mar 15 '21 at 16:55
  • @DippyDog I know. But the question is not clear to me. – callculus42 Mar 15 '21 at 16:55
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    @callculus the question is poorly phrased. I read it as "first you choose a cupcake uniformly at random from $A$ and give it to $B$. Then you choose a cupcake uniformly at random from $B$. What is the probability that this final cupcake is blue?" – lulu Mar 15 '21 at 16:56
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    @DippyDog The prior distribution in $B$ is not directly relevant. All that matters is the distribution in $A$ and the final distribution in $B$. – lulu Mar 15 '21 at 16:57
  • Agree with @lulu, that is what my answer assumes. – Star Bright Mar 15 '21 at 16:57
  • @lulu Thanks. Yes, the was my first idea as well. But I wasn´t sure. DippyDog should rephrase it. My suggestion: A cupcake is finallly chosen from shop B. What is the probability that this cupcake is blue? But I´m still not sure if not something else is meant. – callculus42 Mar 15 '21 at 16:58
  • @callculus And I could have it wrong. But I can't come up with anything else. – lulu Mar 15 '21 at 16:59

1 Answers1

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If we first randomly choose a cupcake from A and transfer it to B, we can say we transfer $\frac{2}{5}$blue + $\frac{3}{5}$ chocolate cupcakes.

If we then randomly choose a cupcake from B, then the probability that the cupcake being blue is $\frac{2.4}{8}$ = 30%.

Star Bright
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