Chapter 5.2, Problem 44E

The following table presents the number of reports of graffiti each of New York’s five over a one-year period. These reports were classified as being open, closed, or pending.

A graffiti report is selected at random. Compute the following probabilities.

The report is open and comes from Brooklyn.

The report is closed or comes from Queens.

The report comes from Manhattan.

The report does not come from Manhattan.

The report is pending.

The report is from the Bronx or Staten Island.

To determine

To find: The probability of selected report is come from Brooklyn.

Answer to Problem 44E

The probability of selected report is come from Brooklyn is $0.2703$

Explanation of Solution

Given information: The data is shown below in table.

  $\boxed{\begin{array}{ccc}\begin{array}{c}\text{Brough }\\ \text{Bronx }\\ \begin{array}{l}\text{Brooklyn}\\ \text{Manhattan}\\ \text{Queens}\\ \text{Staten}\text{Island}\\ \text{Total}\end{array}\end{array}& \begin{array}{c}\text{Open Reports}\\ 1121\\ \begin{array}{l}1170\\ 744\\ 1353\\ 83\\ 4471\end{array}\end{array}& \begin{array}{c}\text{Closed Reports}\\ 1622\\ \begin{array}{l}2706\\ 3380\\ 2043\\ 118\\ 9869\end{array}\end{array}\begin{array}{c}\text{Pending Reports}\\ 80\\ \begin{array}{l}48\\ 25\\ 25\\ 0\\ 178\end{array}\end{array}\begin{array}{c}\text{Total}\\ 2823\\ \begin{array}{l}3924\\ 4149\\ 3421\\ 201\\ 14518\end{array}\end{array}\end{array}}$

Calculation:

The number of reports comes from Brooklyn is $3924$ .

Therefore,

  $P(\text{Brooklyn})=\frac{3924}{14518}=\frac{1962}{7259}=0.2703$

To determine

To find: The probability of selected report is come from Queens.

Answer to Problem 44E

The probability of selected report is come from Queens is $0.2356$

Explanation of Solution

Given information: The data is shown below in table.

  $\boxed{\begin{array}{ccc}\begin{array}{c}\text{Brough }\\ \text{Bronx }\\ \begin{array}{l}\text{Brooklyn}\\ \text{Manhattan}\\ \text{Queens}\\ \text{Staten}\text{Island}\\ \text{Total}\end{array}\end{array}& \begin{array}{c}\text{Open Reports}\\ 1121\\ \begin{array}{l}1170\\ 744\\ 1353\\ 83\\ 4471\end{array}\end{array}& \begin{array}{c}\text{Closed Reports}\\ 1622\\ \begin{array}{l}2706\\ 3380\\ 2043\\ 118\\ 9869\end{array}\end{array}\begin{array}{c}\text{Pending Reports}\\ 80\\ \begin{array}{l}48\\ 25\\ 25\\ 0\\ 178\end{array}\end{array}\begin{array}{c}\text{Total}\\ 2823\\ \begin{array}{l}3924\\ 4149\\ 3421\\ 201\\ 14518\end{array}\end{array}\end{array}}$

Calculation:

The number of reports come from Queens is $3421$ .

So,

  $P(\text{Qeens})=\frac{3421}{14518}=0.2356$

To determine

To find: The probability of selected report is come from Manhattan.

Answer to Problem 44E

The probability of selected report is come from Manhattan is $0.2858$

Explanation of Solution

Given information: The data is shown below in table.

  $\boxed{\begin{array}{ccc}\begin{array}{c}\text{Brough }\\ \text{Bronx }\\ \begin{array}{l}\text{Brooklyn}\\ \text{Manhattan}\\ \text{Queens}\\ \text{Staten}\text{Island}\\ \text{Total}\end{array}\end{array}& \begin{array}{c}\text{Open Reports}\\ 1121\\ \begin{array}{l}1170\\ 744\\ 1353\\ 83\\ 4471\end{array}\end{array}& \begin{array}{c}\text{Closed Reports}\\ 1622\\ \begin{array}{l}2706\\ 3380\\ 2043\\ 118\\ 9869\end{array}\end{array}\begin{array}{c}\text{Pending Reports}\\ 80\\ \begin{array}{l}48\\ 25\\ 25\\ 0\\ 178\end{array}\end{array}\begin{array}{c}\text{Total}\\ 2823\\ \begin{array}{l}3924\\ 4149\\ 3421\\ 201\\ 14518\end{array}\end{array}\end{array}}$

Calculation:

The number of reports come from Manhattan is $4149$ .

Therefore,

  $P(\text{Manhattan})=\frac{4149}{14518}=0.2858$

To determine

To find: The probability of selected report is not come from Manhattan.

Answer to Problem 44E

The probability of selected report is not come from Manhattan is $0.7142$

Explanation of Solution

Given information: The data is shown below in table.

  $\boxed{\begin{array}{ccc}\begin{array}{c}\text{Brough }\\ \text{Bronx }\\ \begin{array}{l}\text{Brooklyn}\\ \text{Manhattan}\\ \text{Queens}\\ \text{Staten}\text{Island}\\ \text{Total}\end{array}\end{array}& \begin{array}{c}\text{Open Reports}\\ 1121\\ \begin{array}{l}1170\\ 744\\ 1353\\ 83\\ 4471\end{array}\end{array}& \begin{array}{c}\text{Closed Reports}\\ 1622\\ \begin{array}{l}2706\\ 3380\\ 2043\\ 118\\ 9869\end{array}\end{array}\begin{array}{c}\text{Pending Reports}\\ 80\\ \begin{array}{l}48\\ 25\\ 25\\ 0\\ 178\end{array}\end{array}\begin{array}{c}\text{Total}\\ 2823\\ \begin{array}{l}3924\\ 4149\\ 3421\\ 201\\ 14518\end{array}\end{array}\end{array}}$

Calculation:

The number of reports come from Manhattan is $4149$ .

It is known that,

  $P(A^c)=1-P(A)$

Therefore,

  $\begin{array}{l}P(\text{Not Manhattan})=1-P(\text{Manhattan})\\ =1-\frac{4149}{14518}=\frac{10369}{14518}=0.7142\end{array}$

To determine

To find: The probability of selected report is pending.

Answer to Problem 44E

The probability of selected report is pending is $0.0123$

Explanation of Solution

Given information: The data is shown below in table.

  $\boxed{\begin{array}{ccc}\begin{array}{c}\text{Brough }\\ \text{Bronx }\\ \begin{array}{l}\text{Brooklyn}\\ \text{Manhattan}\\ \text{Queens}\\ \text{Staten}\text{Island}\\ \text{Total}\end{array}\end{array}& \begin{array}{c}\text{Open Reports}\\ 1121\\ \begin{array}{l}1170\\ 744\\ 1353\\ 83\\ 4471\end{array}\end{array}& \begin{array}{c}\text{Closed Reports}\\ 1622\\ \begin{array}{l}2706\\ 3380\\ 2043\\ 118\\ 9869\end{array}\end{array}\begin{array}{c}\text{Pending Reports}\\ 80\\ \begin{array}{l}48\\ 25\\ 25\\ 0\\ 178\end{array}\end{array}\begin{array}{c}\text{Total}\\ 2823\\ \begin{array}{l}3924\\ 4149\\ 3421\\ 201\\ 14518\end{array}\end{array}\end{array}}$

Calculation:

The number of pending reports is $178$ .

Therefore,

  $P(\text{Pending report})=\frac{178}{14518}=0.0123$

To determine

To find: The probability of selected report is come from Bronx or Staten Island.

Answer to Problem 44E

The probability of selected report is come from Bronx or Staten Island is $0.2083$

Explanation of Solution

Given information: The data is shown below in table.

  $\boxed{\begin{array}{ccc}\begin{array}{c}\text{Brough }\\ \text{Bronx }\\ \begin{array}{l}\text{Brooklyn}\\ \text{Manhattan}\\ \text{Queens}\\ \text{Staten}\text{Island}\\ \text{Total}\end{array}\end{array}& \begin{array}{c}\text{Open Reports}\\ 1121\\ \begin{array}{l}1170\\ 744\\ 1353\\ 83\\ 4471\end{array}\end{array}& \begin{array}{c}\text{Closed Reports}\\ 1622\\ \begin{array}{l}2706\\ 3380\\ 2043\\ 118\\ 9869\end{array}\end{array}\begin{array}{c}\text{Pending Reports}\\ 80\\ \begin{array}{l}48\\ 25\\ 25\\ 0\\ 178\end{array}\end{array}\begin{array}{c}\text{Total}\\ 2823\\ \begin{array}{l}3924\\ 4149\\ 3421\\ 201\\ 14518\end{array}\end{array}\end{array}}$

Calculation:

The number of reports come from Bronx or Staten Island is,

  $2823+201=3024$

Hence,

  $P(\text{Bronx or Staten Island})=\frac{3024}{14518}=0.2083$