Question 1197311
**1. Expected Value of the Die Game**

* **Probability of Winning:** 1/6 (since there's one desired number on a 6-sided die)
* **Probability of Losing:** 5/6
* **Payoff for Winning:** $0 (bet is lost)
* **Payoff for Losing:** -$5 (bet amount)

* **Expected Value:** (Probability of Winning * Payoff for Winning) + (Probability of Losing * Payoff for Losing) 
    = (1/6) * $0 + (5/6) * (-$5) 
    = -$4.17

**Therefore, the expected value of the die game is -$4.17.** This means, on average, you're expected to lose $4.17 per game.

**2. Raffle Ticket Analysis**

* **Total Tickets Sold:** 5000
* **Prizes:**
    * 1 prize of $1000
    * 5 prizes of $200
    * 20 prizes of $50

* **Total Prizes:** 1 + 5 + 20 = 26

* **(a) Probability of Winning a Prize:** 
    * Number of Winning Tickets / Total Tickets 
    * = 26 / 5000 
    * = 0.0052 
    * = 0.52%

* **(b) Expected Value for the Man:**

    * **Total Prize Money:** 
        * ($1000 * 1) + ($200 * 5) + ($50 * 20) = $1000 + $1000 + $1000 = $3000

    * **Expected Winnings:** 
        * (Total Prize Money / Total Tickets) - Cost of Ticket 
        * = ($3000 / 5000) - $5 
        * = $0.60 - $5 
        * = -$4.40

**Therefore, the expected value for the man is -$4.40.** This means, on average, he can expect to lose $4.40 per raffle ticket.