A lottery system has balls numbered 1 to 65 and randomly selects 7 of the lottery balls. There is only one prize of $6,100,000 which is awarded only if a lottery player selects the correct set of 7 lottery balls. If a lottery ticket cost $1.00, what is a lottery player's expected value? How much would the lottery price need to be worth if it were to be a fair game?

To determine the **expected value** of a lottery ticket and to calculate how much the prize should be for the game to be fair, we need to follow a systematic approach.

### Step 1: Number of Possible Combinations
Since the lottery selects 7 balls out of 65 without regard to order, the number of possible combinations is calculated using the combination formula:

\[
C(n, k) = \frac{n!}{k!(n - k)!}
\]

where $ n = 65 $ (the total number of balls) and $ k = 7 $ (the number of balls selected). 

Let's calculate this first.

\[
C(65, 7) = \frac{65!}{7!(65-7)!}
\]

I'll calculate this value now.The total number of possible combinations for selecting 7 balls out of 65 is **696,190,560**.

### Step 2: Probability of Winning
The probability of winning the lottery (selecting the correct 7 balls) is the reciprocal of the total number of combinations:

\[
P(\text{win}) = \frac{1}{696,190,560}
\]

### Step 3: Expected Value Calculation
The expected value (EV) is calculated as the sum of the probabilities of each outcome multiplied by the value of that outcome. In this case, there are two outcomes:

1. **Winning**: The value is \$6,100,000 with a probability of $ \frac{1}{696,190,560} $.
2. **Losing**: The value is -\$1 (the cost of the ticket) with a probability of $ 1 - \frac{1}{696,190,560} $.

So, the expected value $ EV $ is:

\[
EV = \left( \frac{1}{696,190,560} \right) \times 6,100,000 + \left( 1 - \frac{1}{696,190,560} \right) \times (-1)
\]

I'll calculate this now.The expected value (EV) of a lottery ticket is approximately **-\$0.99**. This means that, on average, a player loses \$0.99 for every \$1 spent on a ticket.

### Step 4: Fair Game Calculation
For the game to be fair, the expected value should be 0, meaning the player neither gains nor loses money on average. In this case, we need to adjust the prize such that:

\[
\left( \frac{1}{696,190,560} \right) \times \text{New Prize} + \left( 1 - \frac{1}{696,190,560} \right) \times (-1) = 0
\]

Solving for the new prize:

\[
\text{New Prize} = 696,190,560
\]

Let's confirm the calculation.For the lottery to be a **fair game**, the prize would need to be approximately **\$696,190,560**. This is the value where the expected value of the game becomes zero, meaning a player would neither win nor lose money on average over many plays.

### Summary:
1. The expected value of a lottery ticket is approximately **-\$0.99**.
2. For the game to be fair, the prize should be **\$696,190,560** instead of \$6,100,000.

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Would you like more details or have any other questions?

Here are 5 related questions you might find interesting:
1. How does increasing the number of balls in the lottery affect the expected value?
2. What if the lottery had multiple prize tiers? How would that change the expected value?
3. What is the probability of winning smaller prizes in a typical lottery system with multiple prize levels?
4. How does the expected value of the lottery compare to other common forms of gambling?
5. How would changing the cost of a ticket to \$2 affect the expected value?

**Tip:** Always consider the odds and expected value when participating in any game of chance to understand the risks involved.

This problem involves calculating the expected value of a lottery ticket in a game with 65 numbered balls. It covers the probability of winning, combination theory, and expected value, with step-by-step calculations. The prize needs to be approximately $696,190,560 for the game to be fair.

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