Introduction
- In Probability Theory, Independent events are the important concepts that describe different relationships between events.
- It describes how the occurrence of one event affects the probability of another event.
What are Independent Events?
- Two events are Independent if one event does not affect the probability of the other.
- For example: Flipping heads with a coin has no effect on rolling an even number with some dice, so they are Independent events.
- If the occurrence of one does affect the probability of the other, then it is called Dependent Events.
- Mathematically,
Steps To Solve The Independent Events
To determine whether two events are independent and calculate their probabilities, follow these steps:
1
Identify Events.
2
Use Formula.
3
Calculate the Probability.
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Solved Example
If we flip a fair coin and roll a fair 6-sided die. What’s the probability of getting Tails on the coin and an even number on the die?
SOLUTION
1
Identify Events:
- Event A: Coin lands Tails: $P(A) = \frac{1}{2}$
- Event B: Die shows even number (2, 4 and 6): $P(B) = \frac{3}{6} = \frac{1}{2}$
2
Use Formula:
$$P(A \cap B) = P(A) \cdot P(B)$$
3
Calculate the Probability (Put the values in the formula):
$$P(A \cap B) = \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}$$
Final Answer: $\frac{1}{4}$
Solved Example
A coin is tossed and a die is rolled. What is the probability of getting Heads on the coin and a 4 on the die?
SOLUTION
1
Identify Events:
- Event A: Getting Heads on the coin: $P(A) = \frac{1}{2}$
- Event B: Getting a 4 on the die: $P(B) = \frac{1}{6}$
2
Use Formula:
$$P(A \cap B) = P(A) \cdot P(B)$$
3
Calculate the Probability (Put the values in the formula):
$$P(A \cap B) = \frac{1}{2} \cdot \frac{1}{6} = \frac{1}{12}$$
Final Answer: $\frac{1}{12}$
Solved Example
A bag contains 3 red marbles and 2 blue marbles. You randomly pick a marble, put it back, and then draw again. What is the probability of getting:
- A red marble first
- A blue marble second
SOLUTION
1
Identify Events. Total marbles = 3 red + 2 blue = 5 marbles.
- Event A: First draw is red: $P(A) = \frac{3}{5}$
- Event B: Second Draw is Blue: $P(B) = \frac{2}{5}$
2
Use Formula:
$$P(A \cap B) = P(A) \cdot P(B)$$
3
Calculate the Probability (Put the values in the formula):
$$P(A \cap B) = \frac{3}{5} \cdot \frac{2}{5} = \frac{6}{25}$$
Final Answer: $\frac{6}{25}$
Solved Example
A restaurant serves pizza with 3 topping choices: Pepperoni (P), Mushrooms (M), and Olives (O). Each topping has an independent probability of being selected by a customer.
What is the probability that a customer orders a pizza with both Pepperoni and Mushrooms?
SOLUTION
1
Identify Events:
- Event A: Customer selects Pepperoni: $P(A) = 0.6$
- Event B: Customer selects Mushrooms: $P(B) = 0.4$
2
Use Formula:
$$P(A \cap B) = P(A) \cdot P(B)$$
3
Calculate the Probability (Put the values in the formula):
$$P(A \cap B) = 0.6 \cdot 0.4 = 0.24$$
Final Answer: $0.24$
Solved Example
If roll a fair 6-sided die twice, what is the probability of getting a 5 on the first roll and a 2 on the second roll?
SOLUTION
1
Identify Events:
- Event A: First die roll: $P(5) = \frac{1}{6}$
- Event B: Second die roll: $P(2) = \frac{1}{6}$
2
Use Formula:
$$P(A \cap B) = P(A) \cdot P(B)$$
3
Calculate the Probability (Put the values in the formula):
$$P(A \cap B) = \frac{1}{6} \cdot \frac{1}{6} = \frac{1}{36}$$
Final Answer: $\frac{1}{36}$
