Math Problem Statement

Suppose you toss a coin and will win $5 if it comes up heads. If it comes up tails, you toss again. This time you will receive $10 if it comes up heads. If it comes up tails, toss again. This time you will receive $20 if it is heads and nothing if it comes up tails. What is the mathematical expectation for this game?

Solution

To calculate the expected value of this game, we need to consider all possible outcomes and their probabilities, then multiply each outcome's value by its probability and sum the results.

Step-by-Step Solution:

  1. Define the outcomes and probabilities:

    • On the first toss, the probability of heads (HH) is P(H1)=12P(H_1) = \frac{1}{2}. You win $5.
    • If the first toss is tails (T1T_1), the game continues. The probability of this is P(T1)=12P(T_1) = \frac{1}{2}.
  2. On the second toss, if the first toss was tails:

    • Probability of heads on the second toss (H2H_2) is P(H2)=12P(H_2) = \frac{1}{2}. You win $10.
    • Probability of tails on the second toss (T2T_2) is P(T2)=12P(T_2) = \frac{1}{2}. The game continues.
  3. On the third toss, if both the first and second tosses were tails:

    • Probability of heads on the third toss (H3H_3) is P(H3)=12P(H_3) = \frac{1}{2}. You win $20.
    • Probability of tails on the third toss (T3T_3) is P(T3)=12P(T_3) = \frac{1}{2}. You win $0.

Expected Value Calculation:

E=(value×probability)E = \sum (\text{value} \times \text{probability})

  • From the first toss: E1=5×12=2.5E_1 = 5 \times \frac{1}{2} = 2.5

  • From the second toss (occurs if T1T_1): E2=(10×12)×12=2.5E_2 = \left(10 \times \frac{1}{2}\right) \times \frac{1}{2} = 2.5

  • From the third toss (occurs if T1T_1 and T2T_2): E3=(20×12)×12×12=2.5E_3 = \left(20 \times \frac{1}{2}\right) \times \frac{1}{2} \times \frac{1}{2} = 2.5

Total Expected Value:

E=E1+E2+E3=2.5+2.5+2.5=7.5E = E_1 + E_2 + E_3 = 2.5 + 2.5 + 2.5 = 7.5

Final Answer:

The mathematical expectation for this game is $7.50.


Would you like more details or have questions about this? Here are five related questions to explore further:

  1. How does the expected value change if the payouts on each toss are doubled?
  2. What is the variance and standard deviation of the payouts for this game?
  3. How would the expected value change if the probabilities were altered (e.g., biased coin)?
  4. What happens to the expected value if the game allows infinite retries for tails?
  5. How does the expected value compare to other gambling games, such as dice rolls or card draws?

Tip: Always break down problems into smaller, clear steps to simplify complex calculations!

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Math Problem Analysis

Mathematical Concepts

Probability
Expected Value
Random Variables

Formulas

E = Σ(value × probability)

Theorems

Law of Total Probability
Expected Value of a Discrete Random Variable

Suitable Grade Level

Grades 9-12