Certainly! Let's go through each part of the question step-by-step:
### Part (a): How many different selections are possible?
In the Texas Pick 3 lottery, you select three digits, each ranging from 0 to 9. Since repetition is allowed, each digit can independently be any one of 10 possible values. Therefore, the total number of different combinations of three digits is calculated as:
[tex]\[ \text{Total selections} = 10 \times 10 \times 10 = 1000 \][/tex]
So, the number of different selections possible is 1000.
### Part (b): If you win, what is your net profit?
The net profit is the amount you win minus the cost of the ticket. Here are the given values:
- Prize: [tex]$500
- Cost of the ticket: $[/tex]1
The net profit if you win would be calculated as:
[tex]\[ \text{Net profit} = \text{Prize} - \text{Cost of ticket} = 500 - 1 = 499 \][/tex]
So, if you win, your net profit is [tex]$499.
### Part (c): Find the expected value.
The expected value is a measure of the average outcome if you could play the lottery many times. It can be calculated using the probabilities of winning and losing, along with their respective gains and losses.
1. Probability of winning: Since there is only one correct combination out of 1000 possible selections, the probability of winning is:
\[ \text{Probability of winning} = \frac{1}{1000} \]
2. Probability of losing: If the probability of winning is \(\frac{1}{1000}\), the probability of losing is the complement of this probability:
\[ \text{Probability of losing} = 1 - \frac{1}{1000} = \frac{999}{1000} \]
3. Gains and losses:
- If you win, you gain the net profit of $[/tex]499.
- If you lose, you lose the cost of the ticket, which is [tex]$1.
Now, the expected value can be calculated as:
\[ \text{Expected value} = (\text{Probability of winning} \times \text{Net profit}) - (\text{Probability of losing} \times \text{Cost of ticket}) \]
Substituting the values:
\[ \text{Expected value} = \left(\frac{1}{1000} \times 499\right) - \left(\frac{999}{1000} \times 1\right) \]
\[ \text{Expected value} = \left(0.499\right) - \left(0.999\right) = -0.499 \]
So, the expected value is -$[/tex]0.499.
In summary:
- (a) The number of different selections possible is 1000.
- (b) If you win, your net profit is [tex]$499.
- (c) The expected value is -$[/tex]0.499.
This means, on average, you can expect to lose approximately $0.499 per ticket purchased in the long run.
