Conditional Probability: Dependent Events, Formula, Examples

Dependent Events

Dependent events mean two or more events that occurred after the first or other events has already occurred. For examples there are dependent events in real life:

  • Two times of taking marbles but the first is without replacement, then the second time number of marbles is change.
  • There are three choices of a shirt (A, B and C) to two people but the first person has already taken A, then the second person is only having two choices (B and C).

What is Conditional Probability?

conditional probability is probability of an event that occurred after another event has already occurred. In mathematics, if there are two events (A and B) and event A given when event B has already occurred. It has symbol (A|B).

Formula

Conditional probability of event A and B that is symbolized as (A|B) has formula:

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

It is also can be written as

P(A|B) = P(A ∩ B) / P(B)

Note:

  • P(A)           : probability of event A
  • P(A and B) : probability of the both of events occurred in same time (together)
  • P(B)     : probability of event B

if there are too many events, then the probability is calculating all of possibly combination.

How to solve conditional probability?

There are some steps to finding the conditional probability of event A and B:

  1. Determine probability of event B that has already occurred. It uses probability concept (number of something happen divide by total numbers of outcomes).
  2. Determine the join probability of event A and B. it means the probability of events A and B can happened together divide by possible chance of event B.
  3. Determine the conditional probability by dividing the result of step 2 by step 3.

Examples

1. There are 12 balls in a box that consist of 5 green balls and 7 red balls. The green balls consist of 3 tennis balls and 2 footballs. The red balls consist of 2 tennis balls and 5 footballs. There is a kid that taken one ball out of the box which turn out to be green. What is probability of being its football?

Answer
  • 5 Green balls: 3 tennis & 2 footballs
  • 7 red balls: 2 tennis & 5 footballs
  • event 1 = whether it is green ball or red ball
  • event 2 = whether it is tennis ball or football
  • P(A|B) = probability of ball being green football
  • P(A and B) = join probability = total number of green football / total number of balls = 2/12
  • P(B) = probability of ball being green = total green ball / total number of balls = 5/12

Then, the conditional probability is

P(A|B) = P(A and B) / P(B) = (2/12) / (5/12) = 2/5 = 0,4

So, the probability that the ball is green football is 0,4

2. There are some students in class. 60% of them like mathematics. 45% of them like mathematics and English. What percent of those who like mathematics also like English?

Answer

P(E|M) = probability of students who like mathematics also like English

P(M) = probability students who like mathematics = 0.6

P(E and M) = probability of students who like mathematics and English = 0.45

Then

P(E|M) = P(E and M) / P(M) = 0,45 / 0,6 = 0.75

So, percentage of students who like mathematics also like English = 0.75 = 75%

Read Also

Probability of Independent Events