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Chapter 2: Q36E (page 74)

An academic department with five faculty members narrowed its choice for department head to either candidate A or candidate B. Each member then voted on a slip of paper for one of the candidates. Suppose there are actually three votes for A and two for B. If the slips are selected for tallying in random order, what is the probability that A remains ahead of B throughout the vote count (e.g., this event occurs if the selected ordering is AABAB, but not for ABBAA)?

Short Answer

Expert verified

The probability is \({\rm{0}}{\rm{.2}}{\rm{. }}\) that A remains ahead of B throughout the vote count.

Step by step solution

01

Definition

The term "probability" simply refers to the likelihood of something occurring. We may talk about the probabilities of particular outcomes鈥攈ow likely they are鈥攚hen we're unclear about the result of an event. Statistics is the study of occurrences guided by probability.

02

Finding the probability

We can list all outcomes, see what outcomes are favorable and use

\({\rm{P(A) = }}\frac{{{\rm{ \# favorableoutcomesinA}}}}{{{\rm{ \# outcomesinthesamplespace}}}}{\rm{.}}\)

There are

\(\left( {\begin{aligned}{\rm{5}}\\{\rm{3}}\end{aligned}} \right)\)

ways to select positions for A votes (or equally

\(\left( {\begin{aligned}{\rm{5}}\\{\rm{2}}\end{aligned}} \right)\)

ways to select positions for B votes):

\({\rm{BBAAA,BABAA,BAABA,BAAAB,ABBAA,ABABA,ABAAB,AABBA,AABAB,AAABB}}\)

and only two have \({\rm{A}}\)ahead of \({\rm{B}}\) trough out the vote - \({\rm{AABAB}}\) and\({\rm{AAABB}}\).

All outcomes are equally probable therefore we have that the probability that \({\rm{A}}\) remains ahead of \({\rm{B}}\) trough out the vote is

\(\begin{aligned}{\rm{P(A) = }}\frac{{{\rm{\# favorable outcomes in A }}}}{{{\rm{ \# out comes in the sample space}}}}\\{\rm{ = }}\frac{{\rm{2}}}{{{\rm{10}}}}\\{\rm{ = 0}}{\rm{.2}}{\rm{. }}\end{aligned}\)

Therefore, the probability is \({\rm{0}}{\rm{.2}}{\rm{. }}\)

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Most popular questions from this chapter

An ATM personal identification number (PIN) consists of four digits, each a\({\rm{0, 1, 2, \ldots 8, or 9}}\), in succession.

a. How many different possible PINs are there if there are no restrictions on the choice of digits?

b. According to a representative at the author鈥檚 local branch of Chase Bank, there are in fact restrictions on the choice of digits. The following choices are prohibited: (i) all four digits identical (ii) sequences of consecutive ascending or descending digits, such as \({\rm{6543}}\) (iii) any sequence starting with \({\rm{19}}\) (birth years are too easy to guess). So, if one of the PINs in (a) is randomly selected, what is the probability that it will be a legitimate PIN (that is, not be one of the prohibited sequences)?

c. Someone has stolen an ATM card and knows that the first and last digits of the PIN are \({\rm{8}}\) and\({\rm{1}}\), respectively. He has three tries before the card is retained by the ATM (but does not realize that). So, he randomly selects the \({\rm{2nd and 3rd}}\) digits for the first try, then randomly selects a different pair of digits for the second try, and yet another randomly selected pair of digits for the third try (the individual knows about the restrictions described in (b) so selects only from the legitimate possibilities). What is the probability that the individual gains access to the account?

d. Recalculate the probability in (c) if the first and last digits are \({\rm{1}}\) and \({\rm{1}}\), respectively.

A certain shop repairs both audio and video components. Let A denote the event that the next component brought in for repair is an audio component, and let B be the event that the next component is a compact disc player (so the event B is contained in A). Suppose that \(P\left( A \right) = 0.6\)and\(P\left( B \right) = 0.05\). What is\(P\left( {B|A} \right)\)?

The route used by a certain motorist in commuting to work contains two intersections with traffic signals. The probability that he must stop at the first signal is .4, the analogous probability for the second signal is .5, and the probability that he must stop at at least one of the two signals is .7. What is the probability that he must stop

a. At both signals?

b. At the first signal but not at the second one?

c. At exactly one signal?

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)?

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