Conditional probability

Conditional probability is the probability of an event occurring, assuming that another event has already occurred. $P(B|A)$ is said as the probability of event B given A

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 $D_1$ and the second one $D_2$ . There are 5 ways for $D_1 + D_2 = 6$ and 2 of those ways (distinct) includes a 2. Therefore, our answer is $\frac {2} {5}$ .

Formula

The formula for conditional probability is $P(A \cap B) = P(A) \cdot P(B|A)$ where $P(B|A)$ represents the conditional probability. $P(B|A)$ is also said as the probability of event B occurring given event A occurs. $P(A \cap B)$ is the probability $P(A) \cdot P(B)$ . We can also represent $P(B|A)$ as $P(B|A) = \frac {P(A \cap B)} {P(A)}$

Different Problem

A fair standard die is tossed 3 times. Given that the sum of the first two tosses equals the third, what is the probability that at least one 2 is tossed? (Source AMC)

Solution: The probability that the sum of the first two die is equal to the 3rd die is just another way to also equivalent to the probability that the sum of the numbers on the first two die is less than 7, which is $\frac {5}{12}$ . The probability that a two is rolled and that it meets the first condition is $\frac {2}{9}$ . dividing gets us $\frac {8}{15}$ .

Different approaches

$[asy] size(150); import graph; draw(circle((-1.5,0),3)); real radius = 3; label("$P(A)$",(-1,3),NW); draw(circle((1.5,0),3)); real radius = 3; label("$P(B)$",(1,3),NE); label("$P(A \cap B)$",(0,-3.5)); [/asy]$

In the diagram above, it represents a diagram of the probabilities of A and B occurring, where event B doesn't have to happen, so that event A happens. For example, if we have a group of cats and dogs, and we pick two animals, and the first one is a dog, what is the probability that we pick a cat? Geometrically speaking, $P(A|B)$ would be the ratio of $P(A \cap B)$ to $P(A)$ , since P(B) is 1.

$[asy] size(150); import graph; draw(circle((0,0),3)); real radius = 3; draw(circle((0,-1.5),1)); real radius = 3; [/asy]$

Review Problems

Two fair coins are simultaneously flipped. This is done repeatedly until at least one of the coins comes up heads, at which point the process stops, what is the probability that the other coin also comes up heads on the last flip? (Source HMMT)

The probability that event A occurs is $\frac {3} {4}$ , and the probability that event B occurs is $\frac {2} {3}$ . What are the minimum and maximum possible values of $P(A \cap B)$ ? What are the minimum and maximum possible values of $P(A|B)$ and $P(B|A)$ ? (Source AMC)

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