In real life, we often encounter situations where the result of one event does not affect the result of another event. Imagine you throw a coin and a die simultaneously. Does the result of the coin affect the number that appears on the die? Of course not! These two events are independent.
Independent events are two or more events where the result of one event does not affect the probability of another event. If event A does not affect event B, and vice versa, then both events are independent.
As a simple illustration, think about today's weather and your math exam result tomorrow. Whether it's rainy or sunny today will not affect your exam score (unless you're late because of the rain, but that's another story!). These two events are statistically independent.
The main characteristic of independent events is that the result of one event does not change the probability of another event. In mathematical notation, if A and B are independent events, then:
P(A∣B)=P(A)
This means the probability of A occurring when B has already occurred is the same as the probability of A occurring in general.
To identify whether two events are independent, ask: "Does knowing the result of the first event provide information about the result of the second event?" If the answer is no, then both events are independent.
Since independent events do not affect each other, calculating their joint probability becomes simple. The basic formula for the probability of independent events is:
P(A∩B)=P(A)×P(B)
This formula shows that the probability of both events A and B occurring together equals the multiplication of the probability of each event.
Two dice are rolled simultaneously, one red die and one white die. Find the probability of getting number 2 on the red die and number 5 on the white die.
Solution:
Event A: getting number 2 on the red die
Event B: getting number 5 on the white die
Both events are independent because the result of the red die does not affect the result of the white die.
P(A)=61
P(B)=61
P(A∩B)=P(A)×P(B)=61×61=361
So the probability of getting number 2 on the red die and number 5 on the white die is 361.
In a city, the probability that a fire truck is needed on a particular day is 0.98, while the probability that an ambulance is needed is 0.92. What is the probability that both vehicles are needed on the same day?
These probability numbers represent a scenario of a large city with high emergency activity levels.
Solution:
Event A: fire truck is needed
Event B: ambulance is needed
Both events can be considered independent because the need for a fire truck does not affect the need for an ambulance.
Detailed calculation:
P(A∩B)=P(A)×P(B)
P(A∩B)=0.98×0.92=0.9016
So the probability that both vehicles are needed on the same day is 0.9016 or 90.16%.
Two coins are thrown simultaneously. Find the probability of getting heads on the first coin and tails on the second coin.
A die and a card are drawn from a standard deck. Calculate the probability of getting a prime number on the die and a red card.
In a class, the probability of a student passing mathematics is 0.85 and the probability of passing physics is 0.78. If both subjects are independent, what is the probability that the student passes both subjects?
Three coins are thrown simultaneously. Find the probability of getting exactly two tails.
