Let's solve **Problem 15.3-7** step by step. --- ### **Given:** - You can guess if a coin is **fair** or **two-headed**. - Prior probabilities: \( P(\text{Fair}) = 0.5 \), \( P(\text{Two-headed}) = 0.5 \). - If correct: **Win $5**; if wrong: **Lose $5**. - You can optionally see a demonstration flip before guessing. Let's proceed step by step. --- ## **(a) Decision Analysis Formulation** ### **Actions:** - Guess "Fair" - Guess "Two-headed" ### **States of Nature:** - Coin is Fair (F) - Coin is Two-headed (T) ### **Payoff Table:** | | Coin is Fair (F) | Coin is Two-headed (T) | |-------------------|------------------|------------------------| | Guess Fair | +$5 | -$5 | | Guess Two-headed | -$5 | +$5 | --- ## **(b) Optimal Action without Demonstration Flip** **Expected payoff for each action:** - **Guess Fair:** \( E[\text{Fair}] = 0.5 \times 5 + 0.5 \times (-5) = 2.5 - 2.5 = 0 \) - **Guess Two-headed:** \( E[\text{Two-headed}] = 0.5 \times (-5) + 0.5 \times 5 = -2.5 + 2.5 = 0 \) **Optimal Action:** *Either guess is optimal; expected profit is 0.* --- ## **(c) Find EVPI (Expected Value of Perfect Information)** With perfect information, you always guess correctly. - Probability correct: 1 - Expected profit = $5 (always correct) **EVPI** = Expected profit with perfect info - Expected profit without info EVPI = $5 - $0 = **$5** --- ## **(d) Posterior Distribution after Demonstration Flip** The demonstration is a single flip. Let's compute the posterior using Bayes' Rule. ### **If the demonstration flip is a TAIL:** - **P(Tail | Fair) = 0.5** - **P(Tail | Two-headed) = 0** By Bayes' theorem: \[ P(\text{Fair}|\text{Tail}) = \frac{P(\text{Tail}|\text{Fair}) P(\text{Fair})}{P(\text{Tail})} \] Calculate \(P(\text{Tail})\): \[ P(\text{Tail}) = P(\text{Tail}|\text{Fair}) P(\text{Fair}) + P(\text{Tail}|\text{Two-headed}) P(\text{Two-headed}) \\ = (0.5)(0.5) + (0)(0.5) = 0.25 \] So: \[ P(\text{Fair}|\text{Tail}) = \frac{0.5 \times 0.5}{0.25} = 1 \] \[ P(\text{Two-headed}|\text{Tail}) = \frac{0 \times 0.5}{0.25} = 0 \] ### **If the demonstration flip is a HEAD:** - **P(Head | Fair) = 0.5** - **P(Head | Two-headed) = 1** \[ P(\text{Head}) = (0.5)(0.5) + (1)(0.5) = 0.25 + 0.5 = 0.75 \] \[ P(\text{Fair}|\text{Head}) = \frac{0.5 \times 0.5}{0.75} = \frac{0.25}{0.75} = \frac{1}{3} \] \[ P(\text{Two-headed}|\text{Head}) = \frac{1 \times 0.5}{0.75} = \frac{0.5}{0.75} = \frac{2}{3} \] --- ## **(e) [Excel Check — Skipped in text]** --- ## **(f) Determine Your Optimal Policy** After demonstration flip: - **If Tail:** The coin must be Fair. Guess Fair. **Win $5**. - **If Head:** - Probability Fair: 1/3, Two-headed: 2/3. Expected profit if you guess Fair: \( (1/3) \times 5 + (2/3) \times (-5) = 5/3 - 10/3 = -5/3 \) Expected profit if you guess Two-headed: \( (1/3) \times (-5) + (2/3) \times 5 = -5/3 + 10/3 = 5/3 \) So, **guess Two-headed after seeing Head**. **Summary of Policy:** - If demonstration flip is **Tail**: Guess Fair. - If demonstration flip is **Head**: Guess Two-headed. --- ## **(g) How much should you be willing to pay for the demonstration flip?** Calculate the expected value with the demonstration flip: - **Probability demonstration is Tail:** 0.25 In this case, you win $5. - **Probability demonstration is Head:** 0.75 In this case, you win $5/3$ by guessing Two-headed. So, expected value: \[ E = 0.25 \times 5 + 0.75 \times \left(\frac{5}{3}\right) = 1.25 + 0.75 \times 1.6667 \\ = 1.25 + 1.25 = 2.5 \] **Expected value with demonstration = $2.5** **Expected value without demonstration = $0** **Maximum you should pay = $2.5** --- ## **Final Answers (Summary Table):** **(a)** Actions: Guess Fair/Two-headed; States: Fair/Two-headed; Payoff table provided above. **(b)** Either guess; expected profit = $0. **(c)** EVPI = $5. **(d)** - If see Tail: Posterior (Fair) = 1, (Two-headed) = 0. - If see Head: Posterior (Fair) = 1/3, (Two-headed) = 2/3. **(f)** Optimal policy: - Guess Fair if flip is Tail; - Guess Two-headed if flip is Head. **(g)** Maximum you should pay for demonstration flip: **$2.5**. --- Let me know if you need calculation details or Excel demonstration for part (e)!

Certainly! Let's go through each part of the problem step-by-step with detailed explanations and calculations. --- ## **Problem Recap** - You have a coin that is either **fair** (probability 0.5) or **two-headed** (probability 0.5). - You guess whether it's fair or two-headed. - Correct guess yields **$5**, incorrect guess yields **-$5**. - You can **see a demonstration flip** before guessing (optional). - If you **do not see** the flip, you decide based on prior probabilities. - If you **see a flip**, you update your beliefs and then decide. - Goal: **Maximize expected profit** using Bayes' rule and decision analysis. --- ## **Part (a): Develop the decision analysis formulation** ### **Actions:** - Guess **Fair** (F) - Guess **Two-headed** (T) ### **States of Nature:** - Coin is **Fair** (F) - Coin is **Two-headed** (T) ### **Payoff Table:** | Action | Coin is Fair (F) | Coin is Two-headed (T) | |--------------------|------------------|------------------------| | Guess Fair | +$5 | -$5 | | Guess Two-headed | -$5 | +$5 | ### **Prior Probabilities:** \[ P(F) = 0.5, \quad P(T) = 0.5 \] --- ## **Part (b): Optimal action without seeing a flip** ### **Calculate expected payoffs for each guess:** **Guess Fair:** \[ E[\text{Guess Fair}] = P(F) \times \$5 + P(T) \times (-\$5) = 0.5 \times 5 + 0.5 \times (-5) = 2.5 - 2.5 = \$0 \] **Guess Two-headed:** \[ E[\text{Guess T}] = P(F) \times (-\$5) + P(T) \times \$5 = 0.5 \times (-5) + 0.5 \times 5 = -2.5 + 2.5 = \$0 \] ### **Decision:** - Both guesses yield an expected profit of **$0**. - **Any guess is optimal**; expected value is the same. **Final answer for (b):** *Either guess is optimal; expected profit = $0.* --- ## **Part (c): Expected value of perfect information (EVPI)** ### **Concept:** - With **perfect information**, you always guess correctly: - Expected payoff = **$5** ### **Calculate EVPI:** \[ \text{EVPI} = \text{Expected profit with perfect info} - \text{Expected profit without info} = 5 - 0 = \boxed{\$5} \] --- ## **Part (d): Posterior distribution after demonstration flip** ### **Case 1: Flip is a Tail** **Given:** - \( P(\text{Tail} | F) = 0.5 \) - \( P(\text{Tail} | T) = 0 \) (since two-headed coin always shows Heads) **Prior:** \[ P(F) = 0.5, \quad P(T) = 0.5 \] **Calculate total probability of Tail:** \[ P(\text{Tail}) = P(\text{Tail}|F) P(F) + P(\text{Tail}|T) P(T) = (0.5)(0.5) + (0)(0.5) = 0.25 + 0 = 0.25 \] **Posterior probability coin is fair given Tail:** \[ P(F|\text{Tail}) = \frac{P(\text{Tail}|F) P(F)}{P(\text{Tail})} = \frac{0.5 \times 0.5}{0.25} = \frac{0.25}{0.25} = 1 \] **Posterior probability coin is two-headed given Tail:** \[ P(T|\text{Tail}) = 0 \] **Interpretation:** If you see a tail, **you are certain the coin is fair**. --- ### **Case 2: Flip is a Head** **Given:** - \( P(\text{Head} | F) = 0.5 \) - \( P(\text{Head} | T) = 1 \) **Calculate total probability of Head:** \[ P(\text{Head}) = (0.5)(0.5) + (1)(0.5) = 0.25 + 0.5 = 0.75 \] **Posterior probability coin is fair given Head:** \[ P(F|\text{Head}) = \frac{0.5 \times 0.5}{0.75} = \frac{0.25}{0.75} = \frac{1}{3} \] **Posterior probability coin is two-headed given Head:** \[ P(T|\text{Head}) = \frac{1 \times 0.5}{0.75} = \frac{0.5}{0.75} = \frac{2}{3} \] --- ## **Part (e): Confirm posterior distributions (via Excel or calculations)** *(This is a procedural step. The calculations above are correct.)* --- ## **Part (f): Optimal policy after seeing the flip** ### **If the flip shows Tail:** - **Posterior:** Coin is **certainly fair**. - **Decision:** Guess **Fair** → **Expected payoff = $5**. ### **If the flip shows Head:** - **Posterior probabilities:** \[ P(F) = \frac{1}{3}, \quad P(T) = \frac{2}{3} \] - **Expected payoff if guess Fair:** \[ E = \frac{1}{3} \times 5 + \frac{2}{3} \times (-5) = \frac{5}{3} - \frac{10}{3} = -\frac{5}{3} \approx -\$1.67 \] - **Expected payoff if guess Two-headed:** \[ E = \frac{1}{3} \times (-5) + \frac{2}{3} \times 5 = -\frac{5}{3} + \frac{10}{3} = \frac{5}{3} \approx \$1.67 \] **Decision after seeing Head:** Guess **Two-headed** for an expected profit of **$1.67**. --- ## **Summary of optimal policy:** | Observation | Action | Expected payoff | |--------------|------------------------|-----------------| | Tail | Guess **Fair** | **$5** | | Head | Guess **Two-headed** | **$1.67** | --- ## **Part (g): How much are you willing to pay for the demonstration?** ### **Calculate the expected value of observing the flip:** \[ \text{Expected payoff} = P(\text{Tail}) \times (\text{Payoff if Tail}) + P(\text{Head}) \times (\text{Payoff if Head}) \] - **If Tail:** You will guess **Fair** with certainty, earning **$5**. - **If Head:** You will guess **Two-headed**, earning **$1.67** as shown above. **Probabilities:** \[ P(\text{Tail}) = 0.25, \quad P(\text{Head}) = 0.75 \] **Expected value with observation:** \[ EV = 0.25 \times \$5 + 0.75 \times \$1.67 \approx 0.25 \times 5 + 0.75 \times 1.67 = 1.25 + 1.25 = \$2.50 \] **Expected value without observing:** \[ \$0 \] **Maximum willing to pay:** \[ \$2.50 \] --- # **Final Results Summary:** | Part | Result | |---------|--------------------------------------------------------------| | (a) | Actions: Guess Fair / Guess Two-headed; States: F / T; Payoff table provided | | (b) | Any guess (expected profit = \$0) | | (c) | EVPI = **\$5** | | (d) Tail: \(P(F|Tail)=1\); Head: \(P(F|Head)=1/3\), \(P(T|Head)=2/3\) | | (f) | Tail: Guess Fair; Head: Guess Two-headed | | (g) | Maximum payment for demonstration: **\$2.50** | --- Let me know if you'd like further clarification or assistance with calculations!