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Chapter 2: Q18E (page 65)

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 that at least two bills must be selected to obtain a first $10 bill is 0.67.

Step by step solution

01

Given information

The number of $10 bills that a wallet contains is 5.

The number of $5 bills that a wallet contains is 4.

The number of $1 bills that a wallet contains is 6.

02

Compute the probability

From the provided information, the total number of bills is,

\(5 + 4 + 6 = 15\)

Let X represents the number of trials to select the first $10 bill.

The probability of selecting a $10 bill is \(\frac{5}{{15}}\).

The probability that at least two bills must be selected to obtain a first $10 bill is computed as,

\(\begin{aligned}P\left( {X \ge 2} \right) &= 1 - P\left( {X < 2} \right)\\ &= 1 - P\left( {X = 1} \right)\\ &= 1 - \left( {\frac{5}{{15}}} \right){\left( {1 - \frac{5}{{15}}} \right)^{1 - 1}}\\ &= 1 - \frac{1}{3}\\ &= 1 - 0.33\\ &= 0.67\end{aligned}\)

Therefore, the probability that at least two bills must be selected to obtain a first $10 bill is0.67.

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Most popular questions from this chapter

A mutual fund company offers its customers a varietyof funds: a money-market fund, three different bond funds (short, intermediate, and long-term), two stock funds (moderate and high-risk), and a balanced fund.

Among customers who own shares in just one fund,the percentages of customers in the different funds areas follows:

Money-market 20% High-risk stock 18%

Short bond 15% Moderate-risk stock 25%

Intermediate bond 10% Balanced 7%

Long bond 5%

A customer who owns shares in just one fund is randomlyselected.

a. What is the probability that the selected individualowns shares in the balanced fund?

b. What is the probability that the individual owns shares in a bond fund?

c. What is the probability that the selected individual does not own shares in a stock fund?

Given, \({A_i} = \) {awarded project i}, for \(i = 1, 2, 3\). Use the probabilities given there to compute the following probabilities, and explain in words the meaning of each one.

a. \(P\left( {{A_2}|{A_1}} \right)\)

b. \(P\left( {{A_2} \cap {A_3}|{A_1}} \right)\)

c. \(P\left( {{A_2} \cup {A_3}|{A_1}} \right)\)

d. \(P\left( {{A_1} \cap {A_2} \cap {A_3}|{A_1} \cup {A_2} \cup {A_3}} \right)\)

Show that \(\left( {\begin{array}{*{20}{l}}{\rm{n}}\\{\rm{k}}\end{array}} \right){\rm{ = }}\left( {\begin{array}{*{20}{c}}{\rm{n}}\\{{\rm{n - k}}}\end{array}} \right)\). Give an interpretation involving subsets.

Consider purchasing a system of audio components consisting of a receiver, a pair of speakers, and a \({\rm{CD}}\) player. Let \({{\rm{A}}_{\rm{1}}}\) be the event that the receiver functions properly throughout the warranty period, \({{\rm{A}}_{\rm{2}}}\) be the event that the speakers function properly throughout the warranty period, and \({{\rm{A}}_{\rm{3}}}\) be the event that the \({\rm{CD}}\) player functions properly throughout the warranty period. Suppose that these events are (mutually) independent with \({\rm{P}}\left( {{{\rm{A}}_{\rm{1}}}} \right){\rm{ = }}{\rm{.95}}\), \({\rm{P}}\left( {{{\rm{A}}_{\rm{2}}}} \right){\rm{ = }}{\rm{.98}}\), and \({\rm{P}}\left( {{{\rm{A}}_{\rm{3}}}} \right){\rm{ = }}{\rm{.80}}\).

a. What is the probability that all three components function properly throughout the warranty period?

b. What is the probability that at least one component needs service during the warranty period?

c. What is the probability that all three components need service during the warranty period?

d. What is the probability that only the receiver needs service during the warranty period?

e. What is the probability that exactly one of the three components needs service during the warranty period?

f. What is the probability that all three components function properly throughout the warranty period but that at least one fails within a month after the warranty expires?

Three molecules of type A, three of type B, three of type C, and three of type \({\rm{D}}\) are to be linked together to form a chain molecule. One such chain molecule is ABCDABCDABCD, and another is BCDDAAABDBCC.

a. How many such chain molecules are there? (Hint: If the three were distinguishable from one another鈥� \({\rm{A1,\;A2, A3}}\)鈥攁nd the were also, how many molecules would there be? How is this number reduced when the subscripts are removed from the ?)

b. Suppose a chain molecule of the type described is randomly selected. What is the probability that all three molecules of each type end up next to one another (such as in BBBAAADDDCCC)?
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