Conditional probability and independent events
Simple and Conditional Probability
The main difference between simple probability and conditional probability is whether you have additional information about the event.
Simple Probability
The probability of an event occurring without any extra conditions.
Uses the entire sample space.
Conditional Probability
The probability of an event occurring given that another event has already occurred.
Uses a reduced sample space based on the given condition.
Example 1
Suppose a bag contains 3 red balls and 2 blue balls.
Simple probability:
What is the probability of drawing a red ball?
P(Red) = (n(E)) / (n(S))
P(Red) = 3/5
Conditional probability:
If you already know the ball drawn is not blue, what is the probability it is red?
P(Red | Not Blue) = 3/3
P(Red | Not Blue) = 1
Example 2
A jar contains 10 marbles: 4 green, 3 yellow, 3 red.
Simple probability:
What is the probability of picking a yellow marble?
P(Yellow) = (n(E)) / (n(S))
P(Yellow) = 3/10
Conditional probability:
Suppose you are told that the marble picked is not green.
What is the probability of picking a yellow marble?
P(Yellow | Not Green) = 3/6
P(Yellow | Not Green) = 1/2
In summary, simple probability considers all possible outcomes, while conditional probability recalculates the probability after new information changes the set of possible outcomes.
Calculate conditional probabilities
Example 1: Cards
If a card drawn is known to be a face card (Jack, Queen, or King), what is the probability that it is a King?
Total face cards = 12 (4 Jacks, 4 Queens, 4 Kings)
Number of Kings = 4
P(King | Face Card) = 4/12
P(King | Face Card) = 1/3
Example 2: Students
A class has:
15 boys,
10 girls
Among them:
8 boys play basketball,
4 girls play basketball.
If a randomly selected student is known to play basketball, what is the probability that the student is a boy?
Total basketball players = 8 + 4 = 12
Boys who play basketball = 8
P(Boy | Basketball) = 8/12
P(Boy | Basketball) = 2/3
Example 3: Marbles
A bag contains:
5 red marbles,
7 blue marbles,
8 green marbles.
If the marble drawn is not green, what is the probability that it is blue?
Not green = 5 + 7 = 12
Blue = 7
P(Blue | Not Green) = 7/12
Example 4: Dice
A fair die is rolled. If the result is known to be an even number,
what is the probability that the number is greater than 4?
Even numbers = {2, 4, 6}
Even numbers greater than 4 = {6}
P(Greater then 4 | Even) = 1/3
Example 5: Fruits
A basket contains:
6 apples
5 oranges
4 bananas,
If the selected fruit is not a banana, what is the probability that it is an apple?
Not banana = 6 + 5 = 11
Apples = 6
P(Apple | Not Banana) = 6/11
Determine if two events are independent
Two events are independent if the occurrence of one event does not affect the probability of the other event.
A common way to check independence is to compare:
P(A∩B) and P(A) × P(B)
If P(A∩B) = P(A) × P(B)
then the events are independent.
Otherwise, they are dependent.
Steps to Determine Independence
Step 1: Find P(A) - Probability of event A
Step 2: Find P(B) - Probability of event B
Step 3: Find P(A∩B) - Probability that both happen
Step 4: Compare If P(A∩B) = P(A) x P(B)
Example 1: Die
A fair coin is tossed and a fair die is rolled.
Event A: Getting Heads
Event B: Rolling a 4
Probability of event A
P(A) = 1/2
Probability of event B
P(B) = 1/6
Probability that both happen
P(A∩B) = 1/12
Solve for P(A) × P(B)
P(A) × P(B) = 1/2 x 1/6
P(A) × P(B) = 1/12
Check if P(A∩B) = P(A) × P(B)
Since P(A∩B) = P(A) x P(B) the events are independent.
Example 2: Balls
A bag contains:
5 red balls
3 blue balls
Two balls are drawn without replacement.
Event A: First ball is red.
Event B: Second ball is red.
Probability of event A
P(A) = 5/8
Probability of event B
P(B) = 5/8
Probability that both happen
P(A∩B) = 5/8 x 4/7
P(A∩B) = 20/56
P(A∩B) = 5/14
Solve for P(A) × P(B)
P(A) × P(B) = 5/8 x 5/8
P(A) × P(B) = 25/64
Check if P(A∩B) = P(A) × P(B)
Since P(A∩B) ≠ P(A) x P(B) the events are dependent.
Example 3: Card
A card is drawn from a standard deck.
Event A: The card is a Heart
Event B: The card is a King
Probability of event A
P(A) = 13/52
P(A) =1/4
Probability of event B
P(B) = 4/52
P(B) =1/13
Probability that both happen
P(A∩B) = 1/52
Solve for P(A) × P(B)
P(A) × P(B) = 1/4 x 1/13
P(A) × P(B) =1/52
Check if P(A∩B) = P(A) × P(B)
Since P(A∩B) = P(A) x P(B) the events are independent.
Example 4: Student
A student is chosen from a class.
Event A: The student is a girl.
Event B: The student is a member of the science club.
Given:
P(A) = 0.60
P(B) = 0.40
P(A∩B) = 0.24
Probability of event A
P(A) = 0.60
Probability of event B
P(B) = 0.40
Probability that both happen
P(A∩B) = 0.24
Solve for P(A) × P(B)
P(A) × P(B) = 0.60 x 0.40
P(A) × P(B) = 0.24
Check if P(A∩B) = P(A) × P(B)
Since P(A∩B) = P(A) x P(B) the events are independent.