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What is the Probability of A and B (Independent Events)?
Grade Level:
Class 12
AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics
Definition
What is it?
The probability of two independent events, A and B, happening together is found by multiplying their individual probabilities. 'Independent' means that the outcome of one event does not affect the outcome of the other.
Simple Example
Quick Example
Imagine you toss a coin and then roll a standard six-sided dice. The coin landing on 'Heads' doesn't change the chances of the dice landing on a '4'. These are independent events.
Worked Example
Step-by-Step
What is the probability of getting 'Heads' on a coin toss AND rolling a '6' on a standard dice?
Step 1: Find the probability of getting 'Heads' (Event A). There are 2 possible outcomes (Heads, Tails) and 1 favorable outcome (Heads). So, P(A) = 1/2.
---Step 2: Find the probability of rolling a '6' (Event B). There are 6 possible outcomes (1, 2, 3, 4, 5, 6) and 1 favorable outcome (6). So, P(B) = 1/6.
---Step 3: Since these are independent events, multiply their probabilities to find the probability of both happening. P(A and B) = P(A) * P(B).
---Step 4: P(Heads and 6) = (1/2) * (1/6) = 1/12.
Answer: The probability is 1/12.
Why It Matters
Understanding independent probabilities is super important for many cool fields! In AI/ML, it helps predict if two unrelated factors will both lead to a certain outcome. Engineers use it to calculate the chances of multiple parts of a machine failing at the same time, making our cars and phones safer.
Common Mistakes
MISTAKE: Adding probabilities instead of multiplying them. | CORRECTION: For 'AND' (both events happening), always multiply the probabilities of independent events.
MISTAKE: Confusing independent events with dependent events. | CORRECTION: Remember, for independent events, one outcome absolutely does not change the other's chances. If it does, they are dependent.
MISTAKE: Not reducing fractions to their simplest form. | CORRECTION: Always simplify your final probability fraction (e.g., 2/4 should be 1/2).
Practice Questions
Try It Yourself
QUESTION: A spinner has 4 equal sections: Red, Blue, Green, Yellow. What is the probability of spinning Red AND then flipping a coin and getting Tails? | ANSWER: P(Red) = 1/4. P(Tails) = 1/2. P(Red and Tails) = (1/4) * (1/2) = 1/8.
QUESTION: In a box, there are 5 red marbles and 5 blue marbles. If you pick a marble, put it back, and then pick another, what is the probability of picking a red marble both times? | ANSWER: P(Red first) = 5/10 = 1/2. Since you put it back, the events are independent. P(Red second) = 5/10 = 1/2. P(Red and Red) = (1/2) * (1/2) = 1/4.
QUESTION: You have two bags. Bag 1 has 3 apples and 2 oranges. Bag 2 has 4 apples and 1 orange. What is the probability of picking an apple from Bag 1 AND an orange from Bag 2? | ANSWER: P(Apple from Bag 1) = 3/5. P(Orange from Bag 2) = 1/5. P(Apple from Bag 1 AND Orange from Bag 2) = (3/5) * (1/5) = 3/25.
MCQ
Quick Quiz
If the probability of it raining in Delhi today is 0.3, and the probability of your phone battery dying today is 0.1 (these are independent events), what is the probability that both will happen?
0.4
0.03
0.3
0.1
The Correct Answer Is:
B
For independent events, we multiply their probabilities: 0.3 * 0.1 = 0.03. Option A (0.4) would be for adding probabilities, which is incorrect for 'AND'.
Real World Connection
In the Real World
This concept is used by weather scientists at IMD (India Meteorological Department) when predicting complex weather patterns. For example, if the chance of a clear sky in Bengaluru is 0.7 and the chance of low traffic on a specific highway is 0.6 (and these are considered independent for a simple model), a delivery app like Zepto might use this to estimate the probability of both good conditions for a fast delivery (0.7 * 0.6 = 0.42).
Key Vocabulary
Key Terms
PROBABILITY: The chance of an event happening. | INDEPENDENT EVENTS: Events where the outcome of one does not affect the outcome of the other. | OUTCOME: A possible result of an experiment or event. | FAVORABLE OUTCOME: The specific result you are interested in.
What's Next
What to Learn Next
Next, you should explore 'What is the Probability of A or B (Independent Events)?'. This will teach you how to calculate the chance of at least one of two independent events happening, building directly on what you've learned here!
