Probability Proofs: Show that P(A|B) >= 1 - (1-r)/s

In summary: Your Name]In summary, to show that p(A|B) is greater than or equal to 1-((1-r)/s), we use the fact that p(A&B) is greater than or equal to 0 and rewrite the equation as (s-r)/s, which is a positive number. This proves the inequality as s > r.
  • #1
brendan
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0

Homework Statement



Consider two events A and B such that p(A) = r and p(B) = s with r,s >0 and r + s > 1.


Show that


P(A|B) >or= 1- ( (1-r)/s)







Homework Equations





The Attempt at a Solution



By definition

P(A|B) = P(A & B) / P(B)

We know P(B) = s, so we need an inequality

p(A & B) >= something ... (*)





P(A & B) = P(A) + P(B) - P(A or B)

P(A) and P(B) are given, and we know P(A or B) <= 1

Now (1-r) = p(A'),


So how to I show that p(A & B) is > than p(A'),



regards
Brendan
 
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  • #2


Dear Brendan,

Thank you for your post. To show that p(A&B) is greater than or equal to p(A'), we can use the fact that the probability of any event can never be negative. Therefore, we know that p(A&B) is greater than or equal to 0. We also know that p(A') is equal to 1-p(A), since p(A) and p(A') are complementary events.

Using this information, we can rewrite equation (*) as:

p(A&B) >= 1 - (1-p(A))/s

= (s-p(A))/s

= (s-r)/s

Since r + s > 1, we know that s > r. Therefore, (s-r)/s is a positive number, which means that p(A&B) is greater than or equal to p(A'). This proves the inequality you were trying to show.

I hope this helps. Please let me know if you have any further questions.


 

1. What is the significance of P(A|B) in probability proofs?

P(A|B) represents the conditional probability of event A occurring given that event B has already occurred. It is a measure of the likelihood of A happening after B has occurred.

2. How do we interpret the inequality P(A|B) >= 1 - (1-r)/s in probability proofs?

This inequality states that the conditional probability of A occurring given B is greater than or equal to 1 minus the complement of the probability of B occurring, divided by the sample size. In simpler terms, it means that the likelihood of A happening after B has occurred is at least equal to the probability of B not happening, with respect to the overall sample size.

3. Can you provide an example of how to use P(A|B) in a probability proof?

Sure, let's say we want to determine the probability of flipping a coin twice and getting heads both times. We can represent this event as A and the event of getting heads on the first flip as B. Using the formula P(A|B) = P(A and B) / P(B), we can calculate that P(A|B) = 0.25 / 0.5 = 0.5. This means that given we got heads on the first flip, there is a 50% chance of getting heads on the second flip as well.

4. How does the conditional probability P(A|B) relate to the overall probability of A and B occurring together?

P(A|B) and P(A and B) are related through the formula P(A|B) = P(A and B) / P(B). This means that the probability of both A and B occurring together is equal to the probability of A occurring given B, multiplied by the probability of B occurring.

5. When is the inequality P(A|B) >= 1 - (1-r)/s satisfied in probability proofs?

This inequality is satisfied when the conditional probability of A occurring given B is greater than or equal to the complement of the probability of B occurring, divided by the sample size. In other words, when the likelihood of A happening given B is at least equal to the probability of B not happening, with respect to the overall sample size.

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