Assessment Of Random Variables And Expected Value: Two $20 B
Assessment Random Variables And Expected Value1 Two 20 Bills Three
Assessment: Random Variables and Expected Value 1. Two $20 bills, three $10 bills, four $5 bills, and six $1 bills are placed in a bag. If a bill is chosen at random, what is the expected value for the amount chosen?
2. In a game, you flip a coin twice and record the number of heads that occur. You get 20 points for 2 heads, 5 points for 1 head, and 5 points for no heads. What is the expected value for the number of points you’ll win per turn?
3. Consider a box with gift cards inside. The gift cards amounts are as follows: 10 gift cards with amount of $5 each, 7 gift cards with $10 each, and 5 gift cards with $25 each. Find the expected value for drawing a gift card from the box.
4. "Wheel of Fortune" just got a new wheel! On it, there are 12 slots worth $200, 8 slots worth $400, 5 slots worth $600, 6 slots with no money, 1 slot with $5000, and 1 slot with a car worth $25,000.
a. What is the expected winnings on one turn (cash and prizes)?
b. If you get 12 spins, what are your expected winnings?
5. In a game, you roll a die. If you get a 1, you would win $50. If you roll a 2 or 4, you win $30, and if you roll a 3, 5, or 6, you lose $25. What is the expected value of one roll of the die?
Paper For Above instruction
The concept of expected value is fundamental in probability theory and statistics, providing a measure of the center of the probability distribution of a random variable. It essentially answers the question: "What is the average outcome if the experiment is repeated many times?" This paper explores various applications of expected value by analyzing five different scenarios involving random variables, each demonstrating the calculation of expected value and its implications in real-world contexts.
Scenario 1: Expected Value of a Random Draw from a Set of Bills
In the first scenario, we analyze a collection of bills comprising two $20 bills, three $10 bills, four $5 bills, and six $1 bills. The total number of bills is 2 + 3 + 4 + 6 = 15. To find the expected value of a randomly chosen bill, we multiply each bill denomination by its probability of selection and sum all these products. The probability of selecting any specific bill is 1/15.
Calculating, we have:
Expected value (E) = (2/15)×$20 + (3/15)×$10 + (4/15)×$5 + (6/15)×$1
= (2/15)×20 + (3/15)×10 + (4/15)×5 + (6/15)×1
= (40/15) + (30/15) + (20/15) + (6/15)
Summing these gives: (40 + 30 + 20 + 6) / 15 = 96/15 ≈ $6.40
Thus, the expected value of a randomly chosen bill from this collection is approximately $6.40, indicating on average, a pick yields this amount.
Scenario 2: Expected Points in a Coin Flip Game
The second scenario involves flipping a fair coin twice, with specified points based on the number of heads. The possible outcomes and their probabilities are:
- 2 heads: probability (1/4), points = 20
- 1 head: probability (2/4), points = 5
- 0 heads: probability (1/4), points = 5
Calculations for expected points:
E = (1/4)×20 + (1/2)×5 + (1/4)×5 = 5 + 2.5 + 1.25 = 8.75
Therefore, the expected points per turn in this game are $8.75. This calculation demonstrates how expected value helps predict long-term average winnings in probabilistic games.
Scenario 3: Expected Value of Drawing a Gift Card
The third scenario encompasses a box containing gift cards of different amounts. There are 10 gift cards of $5, 7 of $10, and 5 of $25, totaling 22 cards. The probability and expected contribution of each type are calculated as follows:
Expected value (E) = (10/22)×$5 + (7/22)×$10 + (5/22)×$25
= (10/22)×5 + (7/22)×10 + (5/22)×25
= (50/22) + (70/22) + (125/22) = (50 + 70 + 125) / 22 = 245/22 ≈ $11.14
Thus, the expected value of drawing a gift card from the box is approximately $11.14, indicating the average amount one can expect when randomly selecting a gift card.
Scenario 4: Expected Winnings on a Spin of the Wheel of Fortune
The wheel is divided into various slots: 12 worth $200, 8 worth $400, 5 worth $600, 6 with no money, 1 with $5000, and 1 with a car worth $25000, totaling 12 + 8 + 5 + 6 + 1 + 1 = 33 slots. The expected winnings per spin are calculated by summing the product of each prize and its probability:
Expected value (E) = (12/33)×200 + (8/33)×400 + (5/33)×600 + (6/33)×0 + (1/33)×5000 + (1/33)×25000
= (12×200 + 8×400 + 5×600 + 0 + 5000 + 25000) / 33
= (2400 + 3200 + 3000 + 0 + 5000 + 25000) / 33 = 38600 / 33 ≈ $1169.70
The expected winnings per spin are approximately $1169.70, largely influenced by the chance to win the substantial prizes like the $5000 and the car.
For 12 spins, the total expected winnings are 12×$1169.70 ≈ $14,036.40, illustrating the long-term average earnings over multiple plays.
Scenario 5: Expected Value of a Roll of a Die
In the final scenario, a six-sided die has different payout structures based on the result:
- Roll a 1: win $50
- Roll a 2 or 4: win $30
- Roll a 3, 5, or 6: lose $25
The probabilities for each outcome are:
- P(1) = 1/6
- P(2) = 1/6
- P(3) = 1/6
- P(4) = 1/6
- P(5) = 1/6
- P(6) = 1/6
Calculating expected value:
E = (1/6)×$50 + (1/6)×$30 + (1/6)×$30 + (1/6)×(−$25) + (1/6)×(−$25) + (1/6)×(−$25)
E = (50 + 30 + 30 - 25 - 25 - 25) / 6 = (50 + 30 + 30 - 75) / 6 = 35 / 6 ≈ $5.83
The expected value of a single die roll in this game is approximately $5.83, indicating the average return per roll when considering the payouts and their probabilities.
Conclusion
These diverse scenarios underscore the utility of expected value in assessing potential outcomes across a range of contexts—from simple collections of bills to complex games involving multiple prizes. Understanding how to compute and interpret expected value enables individuals and organizations to make informed decisions, evaluate risks, and optimize strategies in uncertain situations. In all cases, the core principle remains consistent: the expected value is a weighted average of all possible outcomes, with weights corresponding to the probabilities of these outcomes.
References
- Ross, S. M. (2014). A First Course in Probability (9th ed.). Pearson.
- Devore, J. L. (2015). Probability and Statistics for Engineering and the Sciences (8th ed.). Cengage Learning.
- Hogg, R. V., Tanis, E. A., & Zimmerman, D. (2013). Probability and Statistics for Engineers and Scientists (9th ed.). Pearson.
- Walpole, R. E., Myers, R. H., Myers, S. L., & Ye, K. (2012). Probability & Statistics for Engineers & Scientists (9th ed.). Pearson.
- Casella, G., & Berger, R. L. (2002). Statistical Inference (2nd ed.). Duxbury Press.
- Feller, W. (1968). An Introduction to Probability Theory and Its Applications (3rd ed.). Wiley.
- Kennedy, R. (2012). Applied Probability and Statistics (2nd ed.). Wiley.
- Mendenhall, W., Beaver, R. J., & Beaver, J. E. (2013). Introduction to Probability and Statistics (14th ed.). Brooks/Cole.
- Wasserman, L. (2004). All of Statistics: A Concise Course in Statistical Inference. Springer.
- Lehmann, E. L., & Casella, G. (2003). Theory of Point Estimation. Springer.