Independent Events

Two events are independent when the outcome of one event does not affect the result of the other. In simple terms, if one event occurs and does not change the probability of the second event, the events are independent. For example, winning a lottery and going for a picnic have no connection; one does not influence the other, so they are independent events.


Here, we represent the probability of independent events using the formula:

\(P(A ∩ B) = P(A) × P(B)\)

This means that if events A and B are independent, the probability that both events happen is the product of their individual probabilities.


For example,

If event A is rolling the die and getting a 4

\(P(A) = \frac{1}{6}\)

If event B is rolling the coin and getting heads

\(P(A) = \frac{1}{6}\)

Since rolling a die does not affect a coin toss, these events are independent.

\(P(A∩B) = \frac{1}{6} × \frac{1}{2} = \frac{1}{12}\)

So, the probability of getting a four on a die and a head on a coin is \(\frac{1}{12}\).

Probability has two types of events: Independent events and dependent events. Here are a few differences between the two events:

When learning about independent events there are a few certain rules that must be followed such as:

1. Rule of multiplication
This rule is used to find the probability that two independent events co-occur.
For independent events A and B:

\(P(A∩B) = P(A) P(B)\)

For example,

Event A is rolling a die and getting a 6 → \(P(A) = \frac{1}{6}\)

Event B is tossing a coin and getting heads → \(P(B) = \frac{1}{2}\)

Since the die roll does not affect the coin toss, the two events are independent.

P(A ⋂ B) = \(\frac{1}{6}\)\(×\frac{1}{2}\) = \(\frac{1}{12}\)

2. Rule of addition
This rule is used to find the probability that at least one of the two events occurs. It is written as:

\(P(A ∪ B)=P(A)+P(B)-P(A ∩ B)\)

For example,

Let the event A be a student passes math → \(P(A) = 0.7\)

Event B: A student passes science → \(P(B) = 0.6\)

If the events are independent:

\(P(A∩B) = 0.7 × 0.6 = 0.42\)

Now using the addition rule:

\(P(A∪B) = 0.7 + 0.6 – 0.42 = 0.88\)

This means there is an 88% chance the student passes either math, science, or both.

To find the probability of independent events, we use a formula:

                    P(A and B) = \(P(A) × P(B)\)

Where:

P(A and B) is the probability that both events occur.

P(A) is the probability of event A.

P(B) is the probability of event B.

If these statements are true, then A and B are independent events.

Step 1: Calculate the individual probabilities so first, we find the probability of P(A) and then the probability of P(B)

Step 2: Then we calculate the joint probability, which is P(A and B)

Step 3: Determine whether it is independent or not by using the formula P(A and B) = \(P(A) × P(B)\). If it is not true, then events are dependent.

Before using probability formulas, it’s essential to check whether the events are independent or dependent. You can follow these steps.


Step 1: Check whether the events can happen in any order.
If yes, go to step 2. If no, go to step 3.


Step 2: Check if one event changes or affects the outcome of the other.
If yes, go to step 4. If no, go to step 3.


Step 3: If one event does not affect the other, the events are independent. Then, use the formula for independent events.


Step 4: If one event affects the other, the events are dependent. Then, use the formula for dependent events.

Independent events are those where the outcome of one event does not affect the other. Mastering this concept helps in solving probability problems more accurately and confidently.

 

• Remember that independent events don’t impact each other. If something happens to one event, no effect on the other.

• Practice spotting independent and dependent events using simple examples, such as tossing a coin or rolling a die.

• Just keep in mind that for independent events, you multiply their probabilities to find the chance of both happening.

• Try using everyday situations to picture independence, such as putting a card back before drawing again or flipping a coin each time.
  
   
• Parents can create small activities, such as drawing cards with replacement, changing multiple coins, or spinning two spinners, to help children visualize independence.
  
   
• Teachers can give practice worksheets, while parents can encourage children to solve a few problems daily. Regular practice builds confidence and accuracy.

There are many uses of independent events. Let us now see the uses and applications of independent events in different fields:

Coin Tossing in Sports: We use independent events in sports where in football, the coin toss at the start of the game is independent of the match’s outcome. Each toss has a 50% chance of landing heads or tails, no matter what happened before.

 

Rolling Dice in Board Games: We use the concept of independent events in games like Monopoly or Yahtzee, where rolling a die is independent of the previous rolls. The probability of rolling 6 remains 1/6 on each throw.
 

Lottery Draws and Gambling: We use the concept of independent events in lottery draws and gambling, where each lottery ticket has an equal chance of winning, independent of the previous draws. In a roulette game, the ball landing on red is independent of past spins.
 

Weather forecasting: Meteorologists consider each day's weather as an independent event when predicting rainfall or sunshine probabilities.
 

Coin tossing: Each flip of a coin is an independent event, since the result of one toss doesn't affect the outcome of the next.