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Chapter 4: Problem 9

Find the probability.At Least One. In Exercises \(5-12,\) find the probability. Subjects for the next presidential election poll are contacted using telephone numbers in which the last four digits are randomly selected (with replacement). Find the probability that for one such phone number, the last four digits include at least one \(0 .\)

Short Answer

Expert verified
The probability is \(\frac{3439}{10000}\), or 0.3439.

Step by step solution

01

Determine Total Possible Outcomes

Each of the last four digits can be any digit from 0 to 9. Therefore, there are 10 possible choices for each digit. Since there are four digits, the total number of different phone number endings can be calculated as \[ 10^4 = 10000 \]
02

Calculate the Probability of No Zeros

If a phone number has no zeros, each digit can be any of the 9 other digits (1 through 9). The number of combinations for the last four digits without a zero is \[ 9 \times 9 \times 9 \times 9 = 9^4 = 6561 \]
03

Calculate the Probability of At Least One Zero

The probability of a phone number having at least one zero is the complement of the probability of having no zeros. The probability of having no zeros is \[ P(\text{No Zero}) = \frac{9^4}{10^4} = \frac{6561}{10000} \] The probability of at least one zero is therefore \[ P(\text{At Least One Zero}) = 1 - P(\text{No Zero}) = 1 - \frac{6561}{10000} = \frac{3439}{10000} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Theory
Probability theory is the branch of mathematics concerned with analyzing random phenomena. The central idea is to measure the likelihood of different outcomes. In any random experiment, each possible outcome has a probability associated with it. These probabilities must be between 0 and 1, where 0 indicates an impossible event and 1 indicates certainty.
To understand probability, consider an example: If you flip a fair coin, there are two possible outcomes: heads or tails. Since the coin is fair, the probability of getting heads is \(\frac{1}{2}\) and the probability of getting tails is also \(\frac{1}{2}\).
Key concepts in probability theory include:
  • Random Experiments: Processes or actions with uncertain outcomes, like rolling a die or drawing a card.
  • Sample Space (S): The set of all possible outcomes of a random experiment. For a six-sided die, S = {1, 2, 3, 4, 5, 6}.
  • Events: Any subset of the sample space. For example, getting an even number when rolling a die is an event A = {2, 4, 6}.
Complements in Probability
The complement of an event is crucial in probability, as it helps in finding the probability of 'at least one' type of problems.
The complement of an event A, denoted by A', includes all outcomes that are not in A. For instance, if event A is rolling a 2 with a die, A' is rolling a 1, 3, 4, 5, or 6.
  • Probability of Complement: If you know the probability of an event occurring, you can find the probability of it not occurring using the formula: \[ P(A') = 1 - P(A) \]
      Using the given problem:
      • Let A be the event that the last four digits of a phone number include no zeros.
      • The probability of event A (no zeros) is given by \[ P(\text{{No Zeros}}) = \frac{9^4}{10^4} = \frac{6561}{10000} \]

      To find the probability of at least one zero (event A'), we use:
      \[ P(\text{{At Least One Zero}}) = 1 - P(\text{{No Zeros}}) = 1 - \frac{6561}{10000} = \frac{3439}{10000} \]
      Hence, the complement helps in breaking down complex probability scenarios by considering what does not happen and then subtracting from 1.
    • Independent Events
    Independent events are events where the occurrence of one event does not affect the occurrence of another.
    Mathematically, two events A and B are independent if:
    \[ P(A \cap B) = P(A) \times P(B) \]
    For example, flipping a coin and rolling a die are independent events. The outcome of the coin flip doesn't influence the outcome of the die roll.
      Let's delve deeper into independence using the exercise here:
    • Each digit in the phone number is selected independently of others. Choosing a digit does not affect the choice of subsequent digits.
    • This independence implies we can multiply the probabilities of individual digits when determining combinations.
    In our solved problem:
    • The probability of selecting a specific digit for one of the four places is \(\frac{1}{10}\) because there are 10 possible digits (0-9).
    When computing the probability of no zeros, we treated selecting each non-zero digit (1-9) independently:
    • The total number of no-zero combinations for selecting 4 digits: 9 choices for each digit, leads to \(9^4 = 6561\) possible outcomes.
    Thus, understanding independent events aids in easily calculating such probabilities using multiplication of individual probabilities.

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    Most popular questions from this chapter

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