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Chapter 4: Problem 5

Express all probabilities as fractions. A thief steals an ATM card and must randomly guess the correct pin code that consists of four digits (each 0 through 9 ) that must be entered in the correct order. Repetition of digits is allowed. What is the probability of a correct guess on the first try?

Short Answer

Expert verified
\( \frac{1}{10,000} \)

Step by step solution

01

Determine the total number of possible PIN codes

Each digit in the PIN code can be any digit from 0 to 9, which means there are 10 choices for each of the 4 digits. Calculating the total number of possible PIN codes involves multiplying the number of choices for each digit together: \[ 10 \times 10 \times 10 \times 10 = 10^4 = 10,000 \]
02

Identify the number of successful outcomes

There is only one successful outcome, which is guessing the correct PIN code on the first try. Thus, the number of successful outcomes is 1.
03

Calculate the probability of the correct guess

The probability of guessing the correct PIN code is given by the ratio of the number of successful outcomes to the total number of possible outcomes. Therefore, the probability \( P \) is: \[ P = \frac{1 \text{ successful outcome}}{10,000 \text{ possible outcomes}} = \frac{1}{10,000} \]

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Key Concepts

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

probability calculation
Probability in statistics measures how likely an event is to occur. It is calculated using the ratio of the number of successful outcomes to the number of possible outcomes.
In our case, the successful outcome is guessing the correct PIN on the first try. The probability calculation formula is:
\(
\text{Probability} = \frac{\text{Number of Successful Outcomes}}{\text{Total Number of Possible Outcomes}}
\)
To express probabilities as fractions helps in maintaining accuracy and simplicity, which is important in statistical computations.
possible outcomes
Possible outcomes refer to all the different ways an event can occur. For a PIN with four digits where each digit can be any from 0 to 9, we need to calculate the total number of combinations.
Each of the 4 digits in the PIN can independently be any of the 10 digits. This leads to:
\[ 10^4 = 10,000 \]
possible combinations.
Thus, there are 10,000 possible outcomes for the PIN guesses.
successful outcomes
A successful outcome is the specific result we are interested in. In the context of our problem, a successful outcome is guessing the exact PIN correctly on the first try.
Since there is only one correct PIN from the 10,000 possible PINs, we have only one successful outcome.
This is crucial for determining the probability, as all probabilities hinge on the number of successful outcomes relative to possible outcomes.
random guessing
When guessing randomly, each possible outcome has an equal chance of being selected.
In the problem, the thief does not have any clues about the PIN. Therefore, each of the 10,000 possible PIN codes is equally likely.
This concept of random guessing is important because:
  • It ensures that the probability calculation is fair.
  • Each PIN has an equal chance of being the correct one.
Thus, the probability of correctly guessing the PIN on the first try is simply the fraction of one successful outcome over 10,000 possible outcomes:
\[ \frac{1}{10,000} \].

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Most popular questions from this chapter

Express the indicated degree of likelihood as a probability value between 0 and \(1 .\) Based on an Adecco survey of hiring managers who were asked to identify the biggest mistakes that job candidates make during an interview, there is a \(50-50\) chance that they will identify "inappropriate attire."

Express all probabilities as fractions. A Social Security number consists of nine digits in a particular order, and repetition of digits is allowed. After seeing the last four digits printed on a receipt, if you randomly select the other digits, what is the probability of getting the correct Social Security number of the person who was given the receipt?

Find the probability and answer the questions. Each of two parents has the genotype brown/blue, which consists of the pair of alleles that determine eye color, and each parent contributes one of those alleles to a child. Assume that if the child has at least one brown allele, that color will dominate and the eyes will be brown. (The actual determination of eye color is more complicated than that.) a. List the different possible outcomes. Assume that these outcomes are equally likely. b. What is the probability that a child of these parents will have the blue/blue genotype? c. What is the probability that the child will have brown eyes?

Let \(A=\) the event of getting at least 1 defective iPhone when 3 iPhones are randomly selected with replacement from a batch. If \(5 \%\) of the iPhones in a batch are defective and the other \(95 \%\) are all good, which of the following are correct? a. \(P(\bar{A})=(0.95)(0.95)(0.95)=0.857\) b. \(P(A)=1-(0.95)(0.95)(0.95)=0.143\) c. \(P(A)=(0.05)(0.05)(0.05)=0.000125\)

High Fives a. Five "mathletes" celebrate after solving a particularly challenging problem during competition. If each mathlete high fives each other mathlete exactly once, what is the total number of high fives? b. If \(n\) mathletes shake hands with each other exactly once, what is the total number of handshakes? c. How many different ways can five mathletes be seated at a round table? (Assume that if everyone moves to the right, the seating arrangement is the same.) d. How many different ways can \(n\) mathletes be seated at a round table?
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