Difference between revisions of "Conditional probability"
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==Formula== | ==Formula== | ||
| − | The formula for conditional probability is <cmath>P(A \cap B) = P(A) \cdot P(B|A)</cmath> where <math>P(B|A)</math> represents the conditional probability. <math>P(B|A)</math> is also said as the probability of event B occurring given event A occurs. <math>P(A \cap B)</math> is the probability <math>P(A) \cdot P(B)</math>. We can also represent <math>P(A|B)</math> as | + | The formula for conditional probability is <cmath>P(A \cap B) = P(A) \cdot P(B|A)</cmath> where <math>P(B|A)</math> represents the conditional probability. <math>P(B|A)</math> is also said as the probability of event B occurring given event A occurs. <math>P(A \cap B)</math> is the probability <math>P(A) \cdot P(B)</math>. We can also represent <math>P(A|B)</math> as <cmath>P(A|B) = \frac {P(A \cap B)} {P(A)}</cmath> |
==Different cases== | ==Different cases== | ||
What if <math>P(A) = 0</math>? We can easily fix this with a little bit of limits and integrals. We simply just want a random variable conditioned on a random | What if <math>P(A) = 0</math>? We can easily fix this with a little bit of limits and integrals. We simply just want a random variable conditioned on a random | ||
Revision as of 16:59, 30 June 2024
Conditional probability is the probability of an event occurring, assuming that another event has already occurred. 
is said as the probability of event B given A
Contents
Example
Let us say that 2 fair 6 sided dice are rolled and their face up values sum is 6. What is the probability that the face up value of the one dice is 2?
Solution
Let call the first dice 
and the second one 
. There are 5 ways for 
and 2 of those ways (distinct) includes a 2. Therefore, our answer is 
.
Formula
The formula for conditional probability is ![\[P(A \cap B) = P(A) \cdot P(B|A)\]](http://latex.artofproblemsolving.com/6/3/4/6342d7cdd57ec027663742a689ebccfc5429678b.png)
where represents the conditional probability. is also said as the probability of event B occurring given event A occurs. 
is the probability 
. We can also represent 
as ![\[P(A|B) = \frac {P(A \cap B)} {P(A)}\]](http://latex.artofproblemsolving.com/e/1/e/e1e2b8cbd6ed9fadc3b0ed629757084ca540e34b.png)
Different cases
What if 
? We can easily fix this with a little bit of limits and integrals. We simply just want a random variable conditioned on a random
