I need help solving this problem: Subjects for the next presidential election poll are contacted using telephone numbers in which the last 4 digits are randomly selected (with replacement). Find the probability that for one such phone number, the last 4 digits include at least one 0.

MATLAB: An Introduction with Applications
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I need help solving this problem: Subjects for the next presidential election poll are contacted using telephone numbers in which the last 4 digits are randomly selected (with replacement). Find the probability that for one such phone number, the last 4 digits include at least one 0.
**Probability of Randomly Selected Phone Numbers Having at Least One Zero**

In a study to contact subjects for the next presidential election poll, telephone numbers are randomly generated. The numbers consist of four digits (with replacement). The task is to calculate the probability that, for any randomly selected phone number, the last four digits contain at least one '0'.

**Calculation:**
The probability is calculated by determining the opposite scenario—where no digits are zero—and then subtracting from 1.

1. **Determine Total Outcomes:** Each digit can be any number from 0 to 9, so there are \(10^4\) total combinations for four digits.

2. **Calculate No Zeros Scenario:** If no digit is zero, each digit has only 9 possibilities (1 through 9). Hence, there are \(9^4\) combinations where no zero is present.

3. **Calculate Probability of No Zero:** 
   \[
   P(\text{No Zero}) = \frac{9^4}{10^4}
   \]

4. **Calculate Probability of At Least One Zero:**
   \[
   P(\text{At Least One Zero}) = 1 - P(\text{No Zero}) = 1 - \frac{9^4}{10^4}
   \]

**Result:**
Round the final probability to three decimal places for your answer.

(Note: This explanation assumes understanding of basic probability principles, including complementary events. Modify the explanation or provide additional learning materials as necessary based on the learners' proficiency.)
Transcribed Image Text:**Probability of Randomly Selected Phone Numbers Having at Least One Zero** In a study to contact subjects for the next presidential election poll, telephone numbers are randomly generated. The numbers consist of four digits (with replacement). The task is to calculate the probability that, for any randomly selected phone number, the last four digits contain at least one '0'. **Calculation:** The probability is calculated by determining the opposite scenario—where no digits are zero—and then subtracting from 1. 1. **Determine Total Outcomes:** Each digit can be any number from 0 to 9, so there are \(10^4\) total combinations for four digits. 2. **Calculate No Zeros Scenario:** If no digit is zero, each digit has only 9 possibilities (1 through 9). Hence, there are \(9^4\) combinations where no zero is present. 3. **Calculate Probability of No Zero:** \[ P(\text{No Zero}) = \frac{9^4}{10^4} \] 4. **Calculate Probability of At Least One Zero:** \[ P(\text{At Least One Zero}) = 1 - P(\text{No Zero}) = 1 - \frac{9^4}{10^4} \] **Result:** Round the final probability to three decimal places for your answer. (Note: This explanation assumes understanding of basic probability principles, including complementary events. Modify the explanation or provide additional learning materials as necessary based on the learners' proficiency.)
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