Probability proof - what formulas are needed here?

In summary, we are asked to prove that if events A and B are in the same sample space, then if P(A given not B) is greater than P(A), then P(B given A) is less than P(B). To prove this, we use the fact that P(A given B) is equal to the probability of A and B occurring together divided by the probability of B. We cannot assume independence in this case, but we can use the usual definition of independence to prove the statement. Similarly, for the second proof, we use the fact that P(A given B) is equal to the probability of A and B occurring together divided by the probability of B.
  • #1
SavvyAA3
23
0
If events A and B are in the same sample space:
  • .
Proove that if P(A I B') > P(A) then P(B I A) < P(B)

(where B' is the Probability of A given not B)


  • .
Proove that if P(A I B) = P(A) then P(B I A) = P(B)

do we assume independence here so that P(A I B) = [P(A)*P(B)]/ P(B) = P(A) and state that since P(A n B) = P(B n A) that P(B I A) = [P(B)*P(A)] / P(A) = P(B) or is it wrong to assume independence here?
 
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  • #2
For the second proof use the fact that P(A|B)=P(A&B)/P(B) and similarly for the other one. You can't assume independence, but it is easy to see that they are using the usual definition of independence P(A&B)=P(A)P(B).
 
  • #3
Please could you show me the steps you would take
 
  • #4
SavvyAA3 said:
If events A and B are in the same sample space:
  • .
Proove that if P(A I B') > P(A) then P(B I A) < P(B)

(where B' is the Probability of A given not B)

...

assume
[tex]P(A|B')>P(A)[/tex]
then

[tex]\frac{P(A\cap B')}{P(B')}>P(A)[/tex]

[tex]\frac{P(B'|A)P(A)}{P(B')}>P(A)[/tex]

[tex]\frac{P(B'|A)}{P(B')}>1[/tex]

[tex]P(B'|A)>P(B')[/tex]

[tex]1-P(B'|A)<1-P(B')[/tex]

[tex]P(B|A)<P(B)[/tex]

the other one isn't much different
 
  • #5
Thanks soo much!
 

1. What is Probability Proof?

Probability proof is a mathematical method used to determine the likelihood of an event occurring. It involves using mathematical formulas and calculations to determine the probability of a particular outcome.

2. What formulas are needed for Probability Proof?

The specific formulas needed for Probability Proof depend on the type of probability being calculated. Some common formulas include the addition rule, multiplication rule, and Bayes' theorem. Other formulas may be needed depending on the specific scenario being analyzed.

3. How is Probability Proof used in science?

Probability Proof is used in science to analyze and predict the likelihood of various events and outcomes. It is commonly used in fields such as statistics, genetics, and physics to make informed decisions and draw conclusions based on data.

4. What are some common misconceptions about Probability Proof?

One common misconception is that probability proof can determine the exact outcome of an event. In reality, probability only provides an estimate of the likelihood of an outcome. Another misconception is that probability proof can be used to predict rare or unlikely events with certainty, when in fact, it can only provide a probability of occurrence.

5. How can Probability Proof be applied in real-life situations?

Probability Proof can be applied in many real-life situations, such as predicting the likelihood of a disease outbreak, determining the chances of a stock market crash, or calculating the probability of winning a game of chance. It can also be used to make informed decisions in fields such as business, finance, and insurance.

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