About Independent Events in Math
What are Independent Events?
A set of outcomes from an experiment can be referred to as a probability event. These events are classified into three types, ‘dependent events’, ‘independent events’ and ‘mutually exclusive events’.
When the occurrence of one event does not affect the occurrence of the other, the two events are said to be independent events.
If there are two independent events A and B, then
P(A ∩ B) = P(A) × P(B)
Venn Diagram for Independent Events
Real Life Examples
Eating ice-cream and getting soaked in the rain, driving a car and flipping a coin, and so on.
For example: A dice is rolled. If A is the event ‘the outcome is a prime number’ and B is the event ‘the outcome is an even number’.
Sample space (S) = 1, 2, 3, 4, 5, 6
Prime numbers = 1, 2, 3 and 5
∴ P(A)= \(\frac{4}{6}=\frac{2}{3}\)
Even numbers = 2, 4 and 6
∴ P(B) = \(\frac{3}{6}=\frac{1}{2}\)
Also, A and B is the event ‘the outcome is prime number and an even number’.
Hence, P(A ∩ B) = P(A) × P(B)
= \(\frac{2}{3}\times\frac{1}{2}\)
P(A ∩ B) = 13
Rapid Recall
Solved Independent Event Examples
Example 1: State one example of an independent event.
Solution:
Flipping a coin and rolling a dice simultaneously are two separate events. This is because the outcome of the first event, flipping a coin, is separate from the outcome of the second event, rolling a dice.
Example 2: What is the probability of getting a 3 or 5 when a dice is rolled?
Solution:
Sample space (S) = 1, 2, 3, 4, 5 and 6
n(S) = 6
Probability of getting 3 or 5 = 2
Probability of an event = \(\frac{2}{6}=\frac{1}{3}\)
Hence, the probability of getting 3 or 5 is \(\frac{1}{3}\).
Example 3: John and Max were flipping two coins, what is the probability of getting two heads and two tails?
Solution:
Sample space (S) = {HH, HT, TH, TT}
n(S) = 4
Probability of getting two heads and two tails = 2
Probability of an event = \(\frac{2}{4}=\frac{1}{2}\)
Hence, the probability of getting two heads and two tails is \(\frac{1}{2}\).
If two events A and B cannot occur simultaneously, they are said to be mutually exclusive events.
P(A ∩ B) = 0
Example: When a coin is tossed, then the outcome of getting heads or tails are mutually exclusive.
When the outcome of event A affects the outcome of event B, they are said to be dependent events.
Example: Buying 10 lottery tickets and winning the lottery.
The possibility of an event or outcome occurring based on the existence of an earlier event, or outcome is known as conditional probability.
Probability of an event = Number of favorable outcomes / Total number of possible outcomes