To determine the probability of winning with one ticket and the probability of winning if you buy four tickets, let's break down the problem into understandable steps.
### Step 1: Probability of Winning with One Ticket
First, we need to determine the probability of winning with a single ticket. Given that there is 1 winning ticket out of 10,000 possible tickets, the probability of winning with one ticket is:
[tex]\[
\text{Probability of winning with one ticket} = \frac{1}{10,000} = 0.0001
\][/tex]
### Step 2: Probability of Not Winning with One Ticket
Next, we calculate the probability of not winning with a single ticket:
[tex]\[
\text{Probability of not winning with one ticket} = 1 - \text{Probability of winning with one ticket} = 1 - 0.0001 = 0.9999
\][/tex]
### Step 3: Probability of Not Winning with Four Tickets
Now, let's look at the probability of not winning with all four tickets. Since the events are independent (buying one ticket does not affect the outcome of others), we raise the probability of not winning with one ticket to the power of four:
[tex]\[
\text{Probability of not winning with four tickets} = (0.9999)^4
\][/tex]
### Step 4: Probability of Winning with At Least One of the Four Tickets
Finally, we determine the probability of winning with at least one of the four tickets. This is calculated by subtracting the probability of not winning with all four tickets from 1:
[tex]\[
\text{Probability of winning with four tickets} = 1 - \text{Probability of not winning with four tickets} = 1 - (0.9999)^4
\][/tex]
### Numerical Results
Based on these calculations, the results are:
- Probability of winning with one ticket: [tex]\(0.0001\)[/tex]
- Probability of winning with four tickets: [tex]\(1 - (0.9999)^4 = 0.00039994000399989904\)[/tex]
### Answer
From the given options:
A. [tex]\(\frac{1}{10000}\)[/tex]
B. 1
C. [tex]\(\frac{1}{1000}\)[/tex]
D. 1
The correct answer for the probability of winning with one ticket is:
A. [tex]\(\frac{1}{10000}\)[/tex]
For the probability of winning with four tickets, none of the provided options are correct because the calculated probability is approximately [tex]\(0.00039994000399989904\)[/tex], which does not match any of the given choices.
Thus:
- The answer for the probability of winning with one ticket is A. [tex]\(\frac{1}{10000}\)[/tex]
- There is no exact match for the probability of winning with four tickets in the given options.
