Question

Age and senators. According to the congressional directory, the official directory of the U.S Congress prepared by the Joint Committee on printing the age distribution for senators in the U.S Congress as of fall 2013, is as shown in the following table.

Suppose that a U.S senator is selected at random. let

A = event the senator is under 50,

B = event the senator is in his or her 50s,

C = event the senator is in his or her 60s, and

S = event the senator is under 70.

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

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

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

Part(d) Compute P(S), 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) $0.78$

Part (b) $S=(AorBorC)$

Part (c)

$$\begin{gathered} P\left(A\right)=0.11 \\ P\left(B\right)=0.3 \\ P\left(C\right)=0.37 \end{gathered}$$

Part (d)$=0.78$

Step-by-step solution

Part (a) Step 1. Given information.

The following table shows the age distribution of senators in the United States Congress as of Fall 2013.

Age (yr)	No. of senators
Under50	11
50-59	30
60-69	37
70-79	20
80 and over	2

A = event the senator is under 50,

B = event the senator is in his or her 50s,

C = event the senator is in his or her 60s, and

S = event the senator is under 70.

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

Formula that was used:

Probability of event =$\frac{No.offavourableoutcomes}{Totalno.ofoutcomes}$

Total No. of outcomes =$11+30+37+20+2=100$

A favourable number of outcomes for event S =$11+30+37=78$

$P\left(S\right)=\frac{78}{100}=0.78$

Using the table and f/N rule the value of P(s) is$0.78$

Part (b) Step 1. Given information.

Age (yr)	No. of senators
Under50	11
50-59	30
60-69	37
70-79	20
80 and over	2

A = event the senator is under 50,

B = event the senator is in his or her 50s,

C = event the senator is in his or her 60s, and

S = event the senator is under 70.

Part (b) Step 2. To express the occurrence S in terms of the occurrences A, B, and C. 

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

$S=(AorBorC)$

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

Total no. of outcome $=100$

role="math" localid="1651320094416" $$\begin{gathered} P\left(A\right)=\frac{11}{100}=0.11 \\ P\left(B\right)=\frac{30}{100}=0.3 \\ P\left(C\right)=\frac{37}{100}=0.37 \end{gathered}$$

Part (d) Step 1. Given information. 

Age (yr)	No. of senators
Under50	11
50-59	30
60-69	37
70-79	20
80 and over	2

A = event the senator is under 50,

B = event the senator is in his or her 50s,

C = event the senator is in his or her 60s, and

S = event the senator is under 70.

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

$$\begin{gathered} P\left(S\right)=P\left(A\right)+P\left(B\right)+P\left(C\right) \\ 0.11+0.3+0.37=0.78 \end{gathered}$$

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