$Probability of picking a red ball$

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There are 7 balls in a bag. 3 of them are red, and 4 are blue. Your friend randomly removes one of the balls from the bag, but doesn't tell you its color. He asks you to also randomly chose a ball from the bag. What is the probability that you will pick a red ball?

Probability Combinatorics

Savionf

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Let $A$ be the event that your friend removes a red ball and $A^c$ be the event that he/she removes a blue ball. Let also $R$ be the event that you remove a red ball. Then by Bayes's formula
\[P(R)=P(R|A)P(A)+P(R|A^c)P(A^c)=\frac{2}{6}\times \frac{3}{7}+\frac{3}{6}\times \frac{4}{7}=\frac{18}{42}=\frac{3}{7}.\]
As you see, the probability that you will pick a red ball is the same as your friend's chance of removing a red ball! This is because the probability of an event does not change when you do not have any additional information.

Alejandraramirez

$Probability of picking a red ball$

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