Question

1. The Eco Pulse survey from the marketing communications firm Shelton Group asked individuals to indicate things they do that make them feel guilty (Los Angeles Times, August 15, 2012). Based on the survey results, there is a .39 probability that a randomly selected person will feel guilty about wasting food and a .27 probability that a randomly selected person will feel guilty about leaving lights on when not in a room. Moreover, there is a .12 probability that a randomly selected person will feel guilty for both of these reasons.

a. What is the probability that a randomly selected person will feel guilty for either wasting food or leaving lights on when not in a room or both (to 2 decimals)?

b. What is the probability that a randomly selected person will not feel guilty for either of these reasons (to 2 decimals)?

2. Which NCAA college basketball conferences have the higher probability of having a team play in college basketball's national championship game? Over the last 20 years, the Atlantic Coast Conference (ACC) ranks first by having a team in the championship game 10 times. The Southeastern Conference (SEC) ranks second by having a team in the championship game 8 times. However, these two conferences have both had teams in the championship game only one time, when Arkansas (SEC) beat Duke (ACC) 76–70 in 1994 (NCAA website, April 2009). Use these data to estimate the following probabilities.

a. What is the probability the ACC will have a team in the championship game (to 2 decimals)?

b. What is the probability the SEC will have a team in the championship game (to 2 decimals)?

c. What is the probability the ACC and SEC will both have teams in the championship game (to 2 decimals)?

d. What is the probability at least one team from these two conferences will be in the championship game? That is, what is the probability a team from the ACC or SEC will play in the championship game (to 2 decimals)?

e. What is the probability that the championship game will not have a team from one of these two conferences (to 2 decimals)?

Answer 1

Concepts and reason

Probability: The ratio of the number of favorable outcomes to certain event and total number of possible outcomes is called as the probability of an event.

Event: The collection or the set of outcomes in an experiment is called as an event.

Complementary events: If an event A is defined, then complement of event A is not occurring of event A.

Union of two events: The set of the outcomes that belong to either the two events or any of the two events is called as union of two events.

Intersection of two events: The set of the outcomes that belong to both the two events s is called as intersection of two events.

Fundamentals

The probability of an event is defined as,

$\begin{array}{c}\\{\rm{Probability}} = \frac{{{\rm{Number}}\,{\rm{of}}\,{\rm{favorable}}\,{\rm{outcomes}}\,{\rm{for}}\,{\rm{an}}\,{\rm{event}}}}{{{\rm{Total}}\,{\rm{number}}\,{\rm{of}}\,{\rm{outcomes}}}}\\\\ = \frac{{N\left( E \right)}}{{N\left( S \right)}}\\\end{array}$

The probability value for complementary event (A) is, $P\left( {A'} \right) = 1 - P\left( A \right)$

The addition rule for the probability is, $P\left( {A \cup B} \right) = P\left( A \right) + P\left( B \right) - P\left( {A \cap B} \right)$

Some of the other formulas for probability are,

$\begin{array}{c}\\P\left( {A' \cap B} \right) = P\left( B \right) - P\left( {A \cap B} \right)\\\\P\left( {A \cup B} \right)' = 1 - P\left( {A \cup B} \right)\\\end{array}$

$\begin{array}{c}\\P\left( {A' \cup B} \right)' = 1 - P\left( {A' \cup B} \right)\\\\ = 1 - \left[ {P\left( {A'} \right) + P\left( B \right) - P\left( {A' \cap B} \right)} \right]\\\end{array}$

(1.a)

The probability that a randomly selected person will feel guilty for either wasting food or leaving lights on when not in a room or both is obtained as shown below:

From the given information, let events A be the person feels guilty about wasting food and event B be the leaving the light when not in a room respectively.

$P\left( A \right) = 0.39\,\,,\,\,P\left( B \right) = 0.27\,\,,P\left( {A\, \cap \,B} \right) = 0.12$

The required probability is,

$\begin{array}{c}\\P\left( {A \cup B} \right) = P\left( A \right) + P\left( B \right) - P\left( {A \cap B} \right)\\\\ = 0.39 + 0.27 - 0.12\\\\ = 0.66 - 0.12\\\\ = 0.54\\\end{array}$

(1.b)

The probability that a randomly selected person will not feel guilty for either of these reasons is obtained as shown below:

The value for $P\left( {A \cup B} \right)$ is 0.54.

The required probability is,

$\begin{array}{c}\\P\left( {A \cup B} \right)' = 1 - P\left( {A \cup B} \right)\\\\ = 1 - 0.54\\\\ = 0.46\\\end{array}$

(2.a)

The probability of ACC will have a team in the championship game is obtained below:

From the given information, the Atlantic Coast Conference (ACC) ranks first by having a team in the championship game 10 times over the last 20 years.

Number of Atlantic Coast Conference $N\left( A \right)$ is 10.

Total number of years is $N\left( S \right)$ 20.

The required probability is,

$\begin{array}{c}\\P\left( A \right) = \frac{{N\left( A \right)}}{{N\left( S \right)}}\\\\ = \frac{{10}}{{20}}\\\\ = 0.50\\\end{array}$

(2.b)

The probability that the SEC will have a team in the championship game is obtained below:

From the given information, the Southeastern Conference (SEC) ranks second by having a team in the championship game 8 times.

Number of Southeastern Conference (SEC) $N\left( E \right)$ is 8.

Total number of years is $N\left( S \right)$ 20.

The required probability is,

$\begin{array}{c}\\P\left( S \right) = \frac{{N\left( E \right)}}{{N\left( S \right)}}\\\\ = \frac{8}{{20}}\\\\ = 0.40\\\end{array}$

(2.c)

The probability that the ACC and SEC will both have teams in the championship game is obtained below:

From the given information, these two conferences have both had teams in the championship game only one time.

Total number of years is $N\left( S \right) = 20$ .

The required probability is,

$\begin{array}{c}\\P\left( {A \cap S} \right) = \frac{1}{{20}}\\\\ = 0.05\\\end{array}$

(2.d)

The probability that at least one team from these two conferences will be in the championship game is obtained as shown below:

$P\left( {A \cup S} \right) = 0.50\,\,,\,\,P\left( S \right) = 0.40\,\,,P\left( {A\, \cap \,S} \right) = 0.05$

The required probability is,

$\begin{array}{c}\\P\left( {A \cup B} \right) = P\left( A \right) + P\left( B \right) - P\left( {A \cap B} \right)\\\\ = 0.50 + 0.40 - 0.05\\\\ = 0.90 - 0.05\\\\ = 0.85\\\end{array}$

(2.e)

The probability that the championship game will not have a team from one of these tow conferences is obtained as shown below:

The value for $P\left( {A \cup B} \right)$ is 0.85.

The required probability is,

$\begin{array}{c}\\P\left( {A \cup B} \right)' = 1 - P\left( {A \cup B} \right)\\\\ = 1 - 0.85\\\\ = 0.15\\\end{array}$

Ans: Part 1.a

The probability that a randomly selected person will feel guilty for either wasting food or leaving lights on when not in a room or both is 0.54.

Part 1.b

The probability that a randomly selected person will not feel guilty for either of these reasons is 0.46.

Part 2.a

The probability of ACC will have a team in the championship game is 0.50.

Part 2.b

The probability that the SEC will have a team in the championship game is 0.40.

Part 2.c

The probability that the ACC and SEC will both have teams in the championship game is 0.05.

Part 2.d

The probability that at least one team from these two conferences will be in the championship game is 0.85.

Part 2.e

The probability that the championship game will not have a team from one of these two conferences is 0.15.