/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 153 A thief has stolen Roger's autom... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 153

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?

Short Answer

Expert verified
The probability that the thief will find the correct PIN within three tries when he doesn't know the last two digits is 0.03. However, if the thief knows that the third digit is either 1 or 7, with three attempts, the probability of guessing the correct PIN increases to 0.15.

Step by step solution

01

Calculate total possible combinations for the PIN

The thief has to figure out the last two digits of the PIN. Since each digit can vary from 0 to 9, our total possible combinations would be \(10 * 10 = 100 .\) The number 100 comes from ten possible values for the third digit (from 0 to 9) and ten possible values for the fourth digit (from 0 to 9).
02

Calculate probability for the first case

The thief has only three attempts. The thief will therefore have three chances out of the 100 possible combinations. This gives us a probability of \(\frac{3}{100} = 0.03 .\) This is the probability that the thief will find the correct PIN within three tries.
03

Calculate new total combinations

In the second part of the problem, the thief knows that the third digit is either 1 or 7. Therefore, our total combinations are reduced to twenty, as the third digit can take two possible values (1 or 7) and the fourth digit can take ten possible values (from 0 to 9). Hence, the new total combinations are \(2*10=20 .\)
04

Calculate probability for the second case

The probability that the thief guesses the correct PIN within three attempts is now \(\frac{3}{20} = 0.15 .\) This is because the thief still has only three attempts, but now there are only 20 total possible combinations instead of 100.

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Most popular questions from this chapter

Powerball is a game of chance that has generated intense interest because of its large jackpots. To play this game, a player selects five different numbers from 1 through 59, and then picks a Powerball number from 1 through \(39 .\) The lottery organization randomly draws 5 different white balls from 59 balls numbered 1 through 59 , and then randomly picks a Powerball number from 1 through \(39 .\) Note that it is possible for the Powerball number to be the same as one of the first five numbers. a. If a player's first five numbers match the numbers on the five white balls drawn by the lottery organization and the player's Powerball number matches the Powerball number drawn by the lottery organization, the player wins the jackpot. Find the probability that a player who buys one ticket will win the jackpot. (Note that the order in which the five white balls are drawn is unimportant.) b. If a player's first five numbers match the numbers on the five white balls drawn by the lottery organization, the player wins about \(\$ 200,000\). Find the probability that a player who buys one ticket will win this prize.

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