Unexpected Ticket Costs

Solution & Explanation

Understanding the Problem:

You are faced with a decision to buy a second-hand ticket for a sports game with a 20% risk of it being invalid. If invalid, you must buy an official ticket at the stadium for an additional cost. The goal is to determine the expected cost of attending the game and how much money to budget for the event.

Breakdown of Scenarios:

• Scenario 1: The second-hand ticket works (80% probability).

  • Cost: $50 (price of the second-hand ticket).

• Scenario 2: The second-hand ticket does not work (20% probability).

  • Cost: $50 (second-hand ticket) +$70 (official ticket) = $120.

Calculating Expected Cost:

The expected cost is calculated by considering the probability of each scenario and the associated costs:

$E[C] = P(\text{ticket works}) \times \text{Cost if ticket works} + P(\text{ticket fails}) \times \text{Cost if ticket fails}$

Plugging in the values:

$E[C] = 0.8 \times 50 + 0.2 \times 120$ $E[C] = 40 + 24$ $E[C] = 64$

Thus, the expected cost of attending the game is $64.

Budgeting for the Event:

While the expected cost is $64, budgeting for the event involves preparing for the worst-case scenario to ensure you can attend the game regardless of the ticket's validity:

• Worst-case scenario: The second-hand ticket is invalid, and you must buy an official ticket.
  • Total Cost: $120

Therefore, you should budget $120 to cover all possibilities and ensure you can attend the game.

Conclusion:

• Expected Cost: $64
• Budget to Set Aside: $120

This calculation allows you to understand the financial risk associated with buying a second-hand ticket and prepares you for any outcome during the event.