Question

Sales Tax Receipts. The state of Texas maintains records pertaining to the economic development of corporation in the state, From the Economic Development Corporation Report, published by the Texas Comptroller of public Accounts. We obtained the following frequency distribution summarizing the sales tax receipts from the state's Type 4A development corporation during one fiscal year.

A = receipts are less than \(25,000

B = receipts are between \)25,000 and \(49,999

C= receipts are between \)500,000 and \(999,999

D = receipts are at least \)1,000,000, and

R = receipts are either less than \(50,000 or at least \)500,000.

Part (a) Use the table and the f/N rule to find P(R).

Part (b) Express event R in term of event A,B,C and D

Part (c) Determine P(A), P(B),P(C) and P(D).

Part(d) Compute P(R), Using the special addition rule and your answers from part (b) and part(c) Compare your answer with in parts (a)

Short answer

Part (a) $\frac{97}{207}$

Part (b) $R=(AorBorCorD)$

Part (c)

$$\begin{gathered} P\left(A\right)=\frac{25}{207} \\ P\left(B\right)=\frac{23}{207} \\ P\left(C\right)=\frac{17}{207} \\ P\left(D\right)=\frac{32}{207} \end{gathered}$$

Part (d)$\frac{97}{207}$

Step-by-step solution

Part (a) Step 1. Given information.

The following frequency distribution:

Receipts	Frequency
$0-24,999	25
$25,000-49,999	23
$50,000-74,999	21
$75,000-99,999	11
$100,000-199,999	34
$200,000-499,999	44
$500,000-999,999	17
$1,000,000& over	32

A = receipts are less than $25,000,

B = receipts are between $25,000 and $49,999,

C= receipts are between $500,000 and $999,999,

D = receipts are at least $1,000,000, and

R = receipts are either less than $50,000 or at least $500,000.

Part (a) Step 2. To find the value of P(R).

Formula that was used:

Probability of event =

Total No. of outcomes=$25+23+21+11+34+44+17+32=207$

For event R, favourable number of outcomes =$25+23+17+32=97$

$P\left(R\right)=\frac{97}{207}$

Using the table and f/N rule the value of P(R) is$\frac{97}{207}$

Part (b) Step 1. Given information.

Receipts	Frequency
$0-24,999	25
$25,000-49,999	23
$50,000-74,999	21
$75,000-99,999	11
$100,000-199,999	34
$200,000-499,999	44
$500,000-999,999	17
$1,000,000& over	32

A = receipts are less than $25,000,

B = receipts are between $25,000 and $49,999,

C= receipts are between $500,000 and $999,999,

D = receipts are at least $1,000,000, and

R = receipts are either less than $50,000 or at least $500,000.

Part (b) Step 2. In terms of events A, B,C and D, to express the event R.

The event R in terms of events A, B, C, and D can be calculated using the following information:

$R=(AorBorCorD)$

Part (C) Step 1. To determine the worth of the events P(A),P(B), P(C) and P(D)

Total no. of outcome $207$

role="math" localid="1651326488731" $$\begin{gathered} P\left(A\right)=\frac{25}{207} \\ P\left(B\right)=\frac{23}{207} \\ P\left(C\right)=\frac{17}{207} \\ P\left(D\right)=\frac{32}{207} \end{gathered}$$

Part (d) Step 1. Given information. 

Receipts	Frequency
$0-24,999	25
$25,000-49,999	23
$50,000-74,999	21
$75,000-99,999	11
$100,000-199,999	34
$200,000-499,999	44
$500,000-999,999	17
$1,000,000& over	32

A = receipts are less than $25,000,

B = receipts are between $25,000 and $49,999,

C= receipts are between $500,000 and $999,999,

D = receipts are at least $1,000,000, and

R = receipts are either less than $50,000 or at least $500,000.

Part (d) Step 2. The following is the Special Addition rule:

$$\begin{gathered} P\left(R\right)=P\left(A\right)+P\left(B\right)+P\left(C\right)+P\left(D\right) \\ P\left(R\right)=\frac{25}{207}+\frac{23}{207}+\frac{17}{207}+\frac{32}{207}=\frac{97}{207} \end{gathered}$$

Because occurrences A, B, C, and D are mutually exclusive, the result corresponds to portion (a).