Unions and Intersections

[babygirl1501]

Blood Alcohol Level of Victims
A B C Total
AGE 0.00% 0.01-0.09% >or equal to .10%
D= 0-19 142 7 6 155
E= 20-39 47 8 41 96
F 40-59 29 8 77 114
G 60 or over 47 7 35 89
Total 265 30 159 454

Using the table, how many victims were in category described below
A) (A and A')
B) (A' and G')
C) (C or F)
D) (B or G')

would the answer be:
A) 265 and 189
B) 189 and 365
C) 159 or 114
D) 30 or 365

Am I doing this right? What I did for A was add the total of row A to get the 265 and then to find A' is the total of all the other rows? Since a mutually exclusive events means that if one event occurs, the other cannot occur. So event A and its compliment A' are always mutually exclusive.

 

[babygirl1501]

Blood Alcohol Level of Victims
........................ A.................. B................... ...... C .............
AGE ............... 0.00%..... ...... 0.01-0.09%................. >0.10%.........totals
D = 0-19 .......... 142.................. 7...............................6.................155
E= 20=3 ............ 47...................8............................. 41................ 96
F= 40-59 ...........29.....................8.. .................. ........77................114
G= 60 or over ...... 47................... 7...........................35.................89
.......................265....................30.................. ......159 ................454

This was the best I could do with making this chart on the board

 

[soroban]

Hello, babygirl1501!

. . . . . . . \(\displaystyle \text{Blood Alcohol Level of Victims}\)

\(\displaystyle \begin{array}{|c||c|c|c||c|} \hline & A & B & C & \\ \text{Age} & 0.00\% & 0.01-0.09\% & \ge\,0.10\% & \text{Total} \\ \hine \hline \text{D: 0 - 19} & 142 & 7 & 6 & 155 \\ \hline \text{E: 20 - 39} & 47 & 8 & 41 & 96 \\ \hline \text{F: 40 - 59} & 29 & 8 & 77 & 114 \\ \hline \text{G: }\ge\,60 & 47 & 7 & 35 & 89 \\ \hline \text{Total} & 265 & 30 & 159 & 454 \\ \hline \end{array}\)

\(\displaystyle \text{Using the table, how many victims were in category described below?}\)

. . \(\displaystyle (a)\;\;A\cap A'\)

The question asks for the number of victims who are in \(\displaystyle A\) and in not-\(\displaystyle A.\)

The two groups have no people in common.

The answer is 0 (zero).

\(\displaystyle (b)\;\;A' \cap G'\)

This asks for the number of victimes who are not-\(\displaystyle A\) and not-\(\displaystyle G.\)

Count them up . . . There are: .\(\displaystyle 7 + 6 + 8 + 41 + 8 + 77 \:=\: 147\)

\(\displaystyle (c)\;\;C\,\cup\,F\)

This asks for the number of victims who are in \(\displaystyle C\) or in \(\displaystyle F.\)

\(\displaystyle \text{Formula: }\;n(C \cup F) \:=\:n(C) + n(F) - n(C \cap F)\)

\(\displaystyle \text{Therefore: }\:n(C \cap F) \;=\;159 + 114 - 77 \;=\;196\)

\(\displaystyle (d)\;\;B\,\cup \,G'\)

This asks for the number of victimes who are in \(\displaystyle B\) or not-\(\displaystyle G.\)

\(\displaystyle \text{Formula: }\:n(B \cup G') \;=\;n(B) + n(G') - n(B\cap G')\)

\(\displaystyle \text{Therefore: }\:n(B \cup G') \;=\;30 + 365 - 23 \;=\;345\)