[probability theory] simple question about conditional probability

Bayes' Theorem can be used to find the probability of getting a red ball when randomly selecting one from a mix of K red balls and L black balls. The formula is P(red) = (K/(K+L))*p + (L/(K+L))*1, where p is the probability of getting a red ball when randomly selecting one from a mix of one red and one black ball. In summary, the question asks for the probability of selecting a red ball from a mix of K+L balls after previously selecting a red ball and a black ball from a mix of K red balls and L black balls. The use of Venn diagrams can help visualize the problem and Bayes' Theorem can be used to find the
  • #1
rahl___
10
0
Hi all,

I've got this very simple problem:

We have K red balls [or what is the most popular item in combinatorics] and L black balls. If we take one red ball and one black ball and then randomly pick one of them, the probability of getting the red one equils p. We mix all of them, so now we have K+L balls and pick one at random. what is the probability, that it is the red one?

I know it is an elementary problem, but I never really got into that bayes' theorem, which I need to use here, right? I would be grateful for simple and plain explanation.

thanks for your time,
rahl.
 
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  • #2
Venn diagrams are a way easy method for visualizing Bayes' equations.
 
  • #3


Hi Rahl,

Yes, you are correct that this problem involves conditional probability and can be solved using Bayes' theorem. The basic idea behind Bayes' theorem is that it allows us to update our initial probability based on new information. In this case, our initial probability is the probability of picking a red ball given that we have already picked one red and one black ball. This can be written as P(R|RB), where R represents the event of picking a red ball and RB represents the event of picking a red and a black ball.

Using Bayes' theorem, we can rewrite this probability as P(R|RB) = P(RB|R) * P(R) / P(RB), where P(RB|R) represents the probability of picking a red and a black ball given that we have already picked a red ball, P(R) represents the probability of picking a red ball in the first place, and P(RB) represents the overall probability of picking a red and a black ball regardless of the order.

Now, in this problem, we know that P(RB|R) = p, since we are told that the probability of getting a red ball after picking a red and a black ball is equal to p. We also know that P(R) = K / (K+L), since there are K red balls out of a total of K+L balls. Finally, P(RB) = (K+L) / (K+L) = 1, since we are picking one ball out of a total of K+L balls.

Plugging these values into the equation, we get P(R|RB) = p * (K / (K+L)) / 1 = p * (K / (K+L)). So the probability of picking a red ball after we have already picked a red and a black ball is equal to p times the ratio of red balls to total balls.

I hope this helps to clarify the problem for you. Let me know if you have any further questions.

Best,
 

1. What is conditional probability?

Conditional probability is a measure of the likelihood of an event occurring given that another event has already occurred. It is denoted as P(A|B), where A is the event of interest and B is the event that has already occurred.

2. How is conditional probability calculated?

The formula for calculating conditional probability is P(A|B) = P(A and B) / P(B), where P(A and B) is the probability of both events occurring and P(B) is the probability of the event B occurring.

3. What is the difference between conditional probability and joint probability?

Conditional probability focuses on the probability of an event occurring given that another event has already occurred, while joint probability focuses on the probability of both events occurring at the same time.

4. Can conditional probability be applied to real-world situations?

Yes, conditional probability is widely used in various fields such as finance, medicine, and psychology to make informed decisions and predictions based on past events or conditions.

5. What is the importance of conditional probability?

Conditional probability allows us to better understand the relationship between events and make more accurate predictions and decisions. It also helps us in calculating the probability of rare events by considering additional information.

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