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Chapter 2: Problem 18

A wallet contains five \(\$ 10\) bills, four \(\$ 5\) bills, and six \(\$ 1\) bills (nothing larger). If the bills are selected one by one in random order, what is the probability that at least two bills must be selected to obtain a first \(\$ 10\) bill?

Short Answer

Expert verified
The probability is \( \frac{5}{21} \).

Step by step solution

01

Understand the Total Number of Bills

The wallet contains a total of 15 bills. This is calculated as follows: 5 ten-dollar bills, 4 five-dollar bills, and 6 one-dollar bills add up to a total of 15 bills.
02

Determine Probability for the First Bill Not Being $10

The number of non-\(10 bills is 10, consisting of 4 five-dollar and 6 one-dollar bills. The probability that the first bill picked is not a \)10 bill is given by the ratio of non-\(10 bills to total bills:\[P(\text{first bill not \)10}) = \frac{10}{15} = \frac{2}{3}\]
03

Calculate Probability for the Second Bill Being $10 After First Not $10

After picking a non-\(10 bill first, we now have 14 bills left in total, 5 of which are ten-dollar bills. The probability that the second bill selected is a \)10 bill is:\[P(\text{second bill is \(10 | first not \)10}) = \frac{5}{14}\]
04

Calculate Combined Probability

Multiply the probabilities from Step 2 and Step 3 to get the probability of needing at least two bills to get a \(10 bill:\[P(\text{at least 2 bills for \)10}) = \frac{2}{3} \times \frac{5}{14} = \frac{10}{42} = \frac{5}{21}\]
05

Confirm Consistency

Check the steps to ensure the logical flow and arithmetic are correct. We considered all non-$10 bills first and the conditional probability of picking a $10 bill afterward.

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Key Concepts

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

Random Selection
Random selection is an essential concept in probability theory where each item in a set has an equal chance of being chosen. In the exercise given, each bill has an equal likelihood of being selected when drawing from the wallet.
The total number of bills plays a crucial role. Here, we consider the entire set of bills in the wallet, which is 15. Each bill, whether it is a \(10\), \(5\), or \(1\) dollar bill, has the same probability of being drawn on any single attempt.
Here are key considerations for random selection:
  • Each item must have an equal chance of being selected.
  • The process does not favor any item, making it fair and unbiased.
Understanding random selection helps you comprehend various probability scenarios, like the probability of drawing specific bills.
Conditional Probability
Conditional probability measures the likelihood of an event happening given that another event has already occurred. In this exercise, it assesses the chance of the second bill being $10 given that the first is not.
The change in probabilities from drawing one bill affects the probability of subsequent events. Let's break it down further:
  • First, calculate the probability that the first bill drawn is not $10, which changes the context for the next draw.
  • Then, determine the probability of drawing a $10 bill with one fewer bill in the set, noting how the prior event influences this outcome.
Thus, understanding conditional probability is crucial because it allows you to update your probabilities as new information becomes available.
Combinatorics
Combinatorics is the branch of mathematics dealing with combinations and permutations of objects. It helps in calculating probabilities by considering different possible outcomes.
In this example, though the solution primarily uses probability ratios, understanding combinatorics can deepen your comprehension of why those probabilities form. Here's how it relates:
  • Combinatorics helps in determining the number of ways to arrange or select items 鈥� like bills from the wallet.
  • Useful for more complex probability problems where orders and arrangements matter, providing a mathematical foundation for counting possibilities.
Knowing combinatorics will broaden your toolkit for tackling varied probability questions and understanding different outcomes like in this example of selecting bills in the desired probability context.

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Most popular questions from this chapter

Use Venn diagrams to verify the following two relationships for any events \(A\) and \(B\) (these are called De Morgan's laws): a. \((A \cup B)^{\prime}=A^{\prime} \cap B^{\prime}\) b. \((A \cap B)^{\prime}=A^{\prime} \cup B^{\prime}\)

Show that for any three events \(A, B\), and \(C\) with \(P(C)>0\), \(P(A \cup B \mid C)=P(A \mid C)+P(B \mid C)-P(A \cap B \mid C) .\)

The composer Beethoven wrote 9 symphonies, 5 piano concertos (music for piano and orchestra), and 32 piano sonatas (music for solo piano). a. How many ways are there to play first a Beethoven symphony and then a Beethoven piano concerto? b. The manager of a radio station decides that on each successive evening ( 7 days per week), a Beethoven symphony will be played followed by a Beethoven piano concerto followed by a Beethoven piano sonata. For how many years could this policy be continued before exactly the same program would have to be repeated?

Computer keyboard failures can be attributed to electrical defects or mechanical defects. A repair facility currently has 25 failed keyboards, 6 of which have electrical defects and 19 of which have mechanical defects. a. How many ways are there to randomly select 5 of these keyboards for a thorough inspection (without regard to order)? b. In how many ways can a sample of 5 keyboards be selected so that exactly two have an electrical defect? c. If a sample of 5 keyboards is randomly selected, what is the probability that at least 4 of these will have a mechanical defect?

Components of a certain type are shipped to a supplier in batches of ten. Suppose that \(50 \%\) of all such batches contain no defective components, \(30 \%\) contain one defective component, and \(20 \%\) contain two defective components. Two components from a batch are randomly selected and tested. What are the probabilities associated with 0,1 , and 2 defective components being in the batch under each of the following conditions? a. Neither tested component is defective. b. One of the two tested components is defective.
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