Calculate ##P(C|A')## in the given probability problem

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  • #1
chwala
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Homework Statement
See attached ( textbook question).
Relevant Equations
Understanding of conditional probability
1677585086751.png
My interest is on part ##b## only. We know that ##A## and ##B## are independent and not mutually exclusive events therefore,

##P(C)=0.7×0.6=0.42##

##P(C|A')=\dfrac{P(C)-P(A∩C)}{P(A')}=\dfrac{0.42-(0.3×0.42)}{0.7}=\dfrac{0.294}{0.7}=0.42## which is wrong according to textbook solution.

Where is my mistake? cheers.
 

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  • #2
Revisit what the value of Probability of "A intersect C" is.

Are A and C independent?
 
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  • #3
@scottdave wawawawawawa this was a nice one man! Phew. Seen it...
 

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Related to Calculate ##P(C|A')## in the given probability problem

1. What does ##P(C|A')## represent in probability terms?

##P(C|A')## represents the conditional probability of event C occurring given that event A' (the complement of event A) has occurred. It quantifies how likely event C is when event A does not happen.

2. How do you calculate ##P(C|A')## using the definition of conditional probability?

The conditional probability ##P(C|A')## can be calculated using the formula: \[ P(C|A') = \frac{P(C \cap A')}{P(A')} \]where ##P(C \cap A')## is the probability that both events C and A' occur, and ##P(A')## is the probability that event A' occurs.

3. What if the probabilities ##P(C \cap A')## and ##P(A')## are not given directly?

If the probabilities ##P(C \cap A')## and ##P(A')## are not given directly, you may need to use additional information provided in the problem, such as the probabilities of individual events and their complements, to calculate them. For example, if you know ##P(A)## and ##P(C)##, you can use them to find ##P(A') = 1 - P(A)## and potentially ##P(C \cap A')## through other relationships or given data.

4. Can Bayes' Theorem be used to find ##P(C|A')##?

Bayes' Theorem is typically used to find ##P(A|C)## or ##P(A'|C)##, not directly ##P(C|A')##. However, you can use Bayes' Theorem in conjunction with other probability rules to find the necessary probabilities. For example, if you have ##P(A|C)## and ##P(C)##, you can find ##P(C \cap A)## and then use it to find ##P(C \cap A')## as needed.

5. How do you interpret the result of ##P(C|A')## in practical terms?

The result of ##P(C|A')## tells you the likelihood of event C occurring under the condition that event A does not occur. This can be useful in various practical scenarios, such as risk assessment, decision making, and understanding dependencies between events. For instance, if you are analyzing the failure rates of two systems, knowing the probability of one system failing given that the other does not can help in designing more reliable systems.

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