A Prize Of $\$100$, $\$200$, $\$400$, Or $\$800$ Will Be Randomly Awarded To A Game Show Contestant, With The Probabilities Of Winning Each Prize Shown In The Table Below. What Is The Expected Value Of The Prize That

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Introduction

In probability theory, the expected value of a random variable is a measure of the central tendency of the variable's distribution. It represents the long-run average value that the variable is expected to take on when the experiment is repeated many times. In this article, we will calculate the expected value of a prize that will be randomly awarded to a game show contestant.

The Problem

A prize of $100\$100, $200\$200, $400\$400, or $800\$800 will be randomly awarded to a game show contestant, with the probabilities of winning each prize shown in the table below.

Prize Probability
$100\$100 0.20.2
$200\$200 0.30.3
$400\$400 0.20.2
$800\$800 0.30.3

Calculating the Expected Value

To calculate the expected value of the prize, we need to multiply each prize by its corresponding probability and then sum up the results.

Let XX be the random variable representing the prize awarded to the game show contestant. Then, the expected value of XX is given by:

E(X)=∑i=14xipiE(X) = \sum_{i=1}^{4} x_i p_i

where xix_i is the prize and pip_i is the probability of winning the prize.

Step 1: Multiply Each Prize by Its Corresponding Probability

Prize Probability Prize × Probability
$100\$100 0.20.2 $20\$20
$200\$200 0.30.3 $60\$60
$400\$400 0.20.2 $80\$80
$800\$800 0.30.3 $240\$240

Step 2: Sum Up the Results

Now, we need to sum up the results from Step 1 to get the expected value of the prize.

E(X)=$20+$60+$80+$240=$400E(X) = \$20 + \$60 + \$80 + \$240 = \$400

Conclusion

In this article, we calculated the expected value of a prize that will be randomly awarded to a game show contestant. We used the formula for expected value and multiplied each prize by its corresponding probability to get the expected value. The expected value of the prize is $400\$400.

Real-World Applications

The concept of expected value has many real-world applications in finance, insurance, and risk management. For example, in finance, expected value is used to calculate the expected return on investment of a stock or a bond. In insurance, expected value is used to calculate the expected loss of a policyholder. In risk management, expected value is used to calculate the expected cost of a risk.

Example Problems

  1. A lottery ticket costs $2\$2 and has a probability of $10,000\$10,000 of winning the jackpot. What is the expected value of the lottery ticket?
  2. A stock has a probability of $5\$5 of increasing in value and a probability of $3\$3 of decreasing in value. What is the expected value of the stock?

Solutions

  1. The expected value of the lottery ticket is:

E(X)=($10,000×0.0001)−$2=$0.998E(X) = (\$10,000 \times 0.0001) - \$2 = \$0.998

  1. The expected value of the stock is:

E(X)=($5×0.5)−($3×0.5)=$1E(X) = (\$5 \times 0.5) - (\$3 \times 0.5) = \$1

Final Thoughts

Frequently Asked Questions

In this article, we will answer some frequently asked questions about expected value.

Q: What is expected value?

A: Expected value is a measure of the central tendency of a random variable's distribution. It represents the long-run average value that the variable is expected to take on when the experiment is repeated many times.

Q: How is expected value calculated?

A: Expected value is calculated by multiplying each possible outcome by its corresponding probability and then summing up the results.

Q: What are some real-world applications of expected value?

A: Expected value has many real-world applications in finance, insurance, and risk management. For example, in finance, expected value is used to calculate the expected return on investment of a stock or a bond. In insurance, expected value is used to calculate the expected loss of a policyholder. In risk management, expected value is used to calculate the expected cost of a risk.

Q: Can you give an example of how to calculate expected value?

A: Let's say we have a random variable XX that can take on the values 11, 22, or 33 with probabilities 0.20.2, 0.50.5, and 0.30.3, respectively. To calculate the expected value of XX, we would multiply each value by its corresponding probability and then sum up the results:

E(X)=(1×0.2)+(2×0.5)+(3×0.3)=0.2+1+0.9=2.1E(X) = (1 \times 0.2) + (2 \times 0.5) + (3 \times 0.3) = 0.2 + 1 + 0.9 = 2.1

Q: What is the difference between expected value and variance?

A: Expected value and variance are two related but distinct concepts in probability theory. Expected value represents the long-run average value of a random variable, while variance represents the spread or dispersion of the variable's distribution.

Q: Can you give an example of how to use expected value in finance?

A: Let's say we have a stock that has a probability of 0.60.6 of increasing in value by 10%10\% and a probability of 0.40.4 of decreasing in value by 5%5\%. To calculate the expected return on investment of the stock, we would multiply each possible outcome by its corresponding probability and then sum up the results:

E(X)=(0.1×0.6)+(−0.05×0.4)=0.06−0.02=0.04E(X) = (0.1 \times 0.6) + (-0.05 \times 0.4) = 0.06 - 0.02 = 0.04

This means that the expected return on investment of the stock is 4%4\%.

Q: Can you give an example of how to use expected value in insurance?

A: Let's say we have a policyholder who has a probability of 0.80.8 of filing a claim for 10001000 and a probability of 0.20.2 of filing a claim for 50005000. To calculate the expected loss of the policyholder, we would multiply each possible outcome by its corresponding probability and then sum up the results:

E(X)=(1000×0.8)+(5000×0.2)=800+1000=1800E(X) = (1000 \times 0.8) + (5000 \times 0.2) = 800 + 1000 = 1800

This means that the expected loss of the policyholder is 18001800.

Q: Can you give an example of how to use expected value in risk management?

A: Let's say we have a company that has a probability of 0.90.9 of experiencing a loss of 10001000 and a probability of 0.10.1 of experiencing a loss of 50005000. To calculate the expected cost of the risk, we would multiply each possible outcome by its corresponding probability and then sum up the results:

E(X)=(1000×0.9)+(5000×0.1)=900+500=1400E(X) = (1000 \times 0.9) + (5000 \times 0.1) = 900 + 500 = 1400

This means that the expected cost of the risk is 14001400.

Conclusion

In this article, we have answered some frequently asked questions about expected value. We have discussed the concept of expected value, its calculation, and its real-world applications in finance, insurance, and risk management. We have also provided examples of how to use expected value in different contexts. We hope that this article has provided a clear understanding of the concept of expected value and its applications.