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What is the Expected Value of a Discrete Random Variable?
Grade Level:
Class 12
AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics
Definition
What is it?
The Expected Value (or Expectation) of a Discrete Random Variable is the average outcome we would expect if we repeated an experiment many, many times. It's like the 'long-run' average result, calculated by multiplying each possible value by its probability and adding them up.
Simple Example
Quick Example
Imagine you play a game where you roll a standard six-sided dice. If you roll a 6, you win Rs. 10. Otherwise, you lose Rs. 2. What's the expected amount of money you'll win or lose per roll? The Expected Value helps us figure out this average gain/loss over many rolls.
Worked Example
Step-by-Step
Let's calculate the expected number of heads if you flip a fair coin twice.
1. First, list all possible outcomes and the number of heads for each:
- HH (2 heads)
- HT (1 head)
- TH (1 head)
- TT (0 heads)
2. Next, find the probability of each number of heads:
- P(X=0 heads) = P(TT) = 1/4
- P(X=1 head) = P(HT or TH) = 2/4 = 1/2
- P(X=2 heads) = P(HH) = 1/4
3. Now, multiply each possible number of heads (X) by its probability P(X):
- For 0 heads: 0 * (1/4) = 0
- For 1 head: 1 * (1/2) = 1/2
- For 2 heads: 2 * (1/4) = 2/4 = 1/2
4. Finally, add these values together:
- Expected Value E(X) = 0 + 1/2 + 1/2 = 1
So, the Expected Value is 1. This means if you flip a coin twice many times, on average, you would expect to get 1 head.
Why It Matters
Understanding Expected Value is super important in fields like Finance, AI, and even game design. Financial experts use it to decide if an investment is worth the risk, while engineers use it to predict how long a machine might last. It helps make smarter decisions in situations involving uncertainty.
Common Mistakes
MISTAKE: Assuming Expected Value is one of the actual possible outcomes. | CORRECTION: Expected Value is an average and might not be a value the variable can actually take. For example, the expected number of children per family might be 2.3, but no family has 2.3 children.
MISTAKE: Forgetting to multiply each value by its probability. | CORRECTION: The formula E(X) = Sum [x * P(X=x)] explicitly requires multiplying each outcome by its likelihood before summing them up. Just adding outcomes will give a wrong answer.
MISTAKE: Not ensuring probabilities sum to 1. | CORRECTION: Before calculating, always check that the sum of all probabilities for the discrete random variable equals 1. If not, your probabilities are incorrect, and your Expected Value will be wrong.
Practice Questions
Try It Yourself
QUESTION: A lucky draw ticket costs Rs. 50. You can win Rs. 1000 with a probability of 0.01, or nothing with a probability of 0.99. What is your expected gain or loss? | ANSWER: Expected Winnings = (1000 * 0.01) + (0 * 0.99) = 10. Expected Gain/Loss = 10 (winnings) - 50 (cost) = -Rs. 40. So, an expected loss of Rs. 40.
QUESTION: In a game, you roll a single die. If you roll an even number, you win Rs. 6. If you roll an odd number, you lose Rs. 3. What is the expected value of your winnings per roll? | ANSWER: Possible outcomes: Even (2, 4, 6) and Odd (1, 3, 5). P(Win Rs. 6) = 3/6 = 1/2. P(Lose Rs. 3) = 3/6 = 1/2. Expected Value = (6 * 1/2) + (-3 * 1/2) = 3 - 1.5 = Rs. 1.5.
QUESTION: A small shop sells 0, 1, or 2 mobile phone covers per day with probabilities 0.3, 0.5, and 0.2 respectively. Each cover costs Rs. 100 and sells for Rs. 150. What is the expected daily profit from selling mobile covers? | ANSWER: Profit for 0 sales = 0. Profit for 1 sale = 150 - 100 = 50. Profit for 2 sales = 2 * (150 - 100) = 100. Expected Profit = (0 * 0.3) + (50 * 0.5) + (100 * 0.2) = 0 + 25 + 20 = Rs. 45.
MCQ
Quick Quiz
A bag contains 3 red balls and 2 blue balls. You draw one ball. If it's red, you win Rs. 20. If it's blue, you lose Rs. 30. What is the expected value of your winnings?
Rs. 10
Rs. 0
Rs. 4
-Rs. 6
The Correct Answer Is:
B
Probability of red = 3/5. Probability of blue = 2/5. Expected Value = (20 * 3/5) + (-30 * 2/5) = 12 - 12 = 0. So, the expected winnings are Rs. 0.
Real World Connection
In the Real World
Cricket analysts use Expected Value to predict how many runs a team might score from a particular over, helping captains make strategic decisions. Similarly, in FinTech, companies use it to evaluate investment risks, like whether to fund a new startup, by calculating the expected return versus potential loss.
Key Vocabulary
Key Terms
DISCRETE RANDOM VARIABLE: A variable whose value can only be a specific, countable number, like the number of heads in coin flips. | PROBABILITY: The chance of a specific event happening. | OUTCOME: A possible result of an experiment. | EXPECTATION: Another name for Expected Value. | WEIGHTED AVERAGE: An average where each value contributes differently based on its importance or probability.
What's Next
What to Learn Next
Great job understanding Expected Value! Next, you can explore the 'Variance of a Discrete Random Variable'. Variance helps us understand how spread out or 'risky' the outcomes are around the expected value, which is crucial for making even smarter decisions.
