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A thief has stolen Roger's automatic teller machine (ATM) card. The card has a four-digit personal identification number (PIN). The thief knows that the first two digits are 3 and 5 , but he does not know the last two digits. Thus, the PIN could be any number from 3500 to \(3599 .\) To protect the customer, the automatic teller machine will not allow more than three unsuccessful attempts to enter the PIN. After the third wrong PIN, the machine keeps the card and allows no further attempts. a. What is the probability that the thief will find the correct PIN within three tries? (Assume that the thief will not try the same wrong PIN twice.) b. If the thief knew that the first two digits were 3 and 5 and that the third digit was either 1 or 7 , what is the probability of the thief guessing the correct PIN in three attempts?

Step by step solution

01

Determine the Number of Options for the PIN

The thief knows the first two digits (3 and 5) but does not know the last two. There are 10 possibilities for each unknown digit (0 through 9), so there are \(10 \times 10 = 100\) possible combinations for the last two digits.

02

Calculate the Probability for Part a

The probability that the thief guesses the correct PIN in any single try is \(1/100\). Since the thief has three trials and will not repeat a previous wrong PIN, the total probability is the sum of the individual probabilities, which is \(3/100\) or 0.03.

03

Consider the Additional Information for Part b

If the thief knew the third digit was either a 1 or a 7, this reduces the number of possible combinations for the PIN. Now there are only \(2 \times 10 = 20\) possible combinations.

04

Calculate the Probability for Part b

This situation is similar to part a but with fewer possible PINs. The probability that the thief guesses the correct PIN in a single try is \(1/20\). Given three trials without repetition, the total probability is the sum of the individual probabilities, which is \(3/20\) or 0.15.

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Key Concepts

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

Combinatorics

Combinatorics is a branch of mathematics that deals with counting, arrangement, and combination of objects. In our scenario, it helps in understanding how many possible combinations one has for the PIN digits. The thief knew that the first two digits of the PIN were fixed as 3 and 5. However, the remaining two digits could be anything from 0 to 9.

• There are 10 possible choices for each unknown digit, making it a combinatory problem where you multiply the options for one position by the options for the other.
• This leads to a total of \(10 \times 10 = 100\) possible combinations for the last two digits of the PIN.

Combinatorics makes these counting problems easy to tackle. Knowing how to calculate possibilities allows us to then use this information for probability calculations.

Probability Calculation

Probability calculation is about determining how likely an event is to occur. In our problem, we want to determine the probability that the thief guesses the correct PIN.The probability \(P\) of an event is calculated as:\[P( ext{Event}) = \frac{\text{Number of Successful Outcomes}}{\text{Total Number of Possible Outcomes}}\]In part a of our problem, the successful outcome is the thief guessing the correct PIN out of 100 possible options. Hence, the probability of correctly guessing on one try is \(\frac{1}{100}\).

• With three attempts without repetition, the probability of success increases as he has more chances to be correct. Thus, the overall probability becomes:\[\frac{3}{100} = 0.03\]

Probability calculation allows us to understand the chances of an event happening, a must-have skill when dealing with randomness in situations like this.

Conditional Probability

Conditional probability is the likelihood of an event or outcome occurring, based on the occurrence of a previous event. Part b of our problem introduces a condition: the thief knows the third digit is either a 1 or a 7.

• This reduces the potential combinations from 100 to 20 because now each combination is limited to just these two choices for the third digit and any number for the fourth.

The formula for calculating conditional probability is:\[P(A|B) = \frac{P(A \cap B)}{P(B)}\]However, in our simplified case, since we've been given specific information, it reduces the number of total outcomes directly to 20 possible PINs.

• The probability of the thief guessing correctly in one try with this knowledge is \(\frac{1}{20}\).
• With three attempts, it becomes \( \frac{3}{20} = 0.15 \).

Conditional probability is particularly useful when additional information is available, helping us to refine our predictions and calculations in a given scenario.