a To complete the table we need to remember that the frequencies in each row and column add up to the corresponding values from the row or column Totals.
B
b Consider the completed table. Note that the condition of a person being vaccinated limits the relative frequency of total possible outcomes to 60 %=0.6.
C
c Consider the completed table. Note that the condition of a person not being vaccinated limits the relative frequency of total possible outcomes to 40 %=0.4.
A
a
Got the Flu
Did Not Get the Flu
Totals
Vaccinated
6 %
54 %
60 %
Not Vaccinated
9 %
31 %
40 %
Totals
15 %
85 %
100 %
B
b 10 %
C
c 22.5 %
Practice makes perfect
a We are given a table with a relative frequency distribution for healthy people under the age of 65. First we want to complete it. Consider the given table.
Got the Flu
Did Not Get the Flu
Totals
Vaccinated
54 %
60 %
Not Vaccinated
Totals
15 %
100 %
To complete the table we need to remember that the frequencies in each row and column add up to the corresponding values from the row or column Totals. Considering this fact, we can start with finding the relative frequency of vaccinated people who got the flu. We will use the fact that this percentage and 54 % add up to 60 %.
Vaccinated, flu: 60 %-54 %= 6 %
Consider now the column Totals. The percentage of not vaccinated people and the percentage of vaccinated people, 60 %, add up to 100 %. Also in the row Totals the percentage of people who got the flu, 15 %, and the percentage of people who did not get the flu add up to 100 %. We can use these facts to find another two missing relative frequencies.
Not vaccinated:& 100 %-60 %= 40 %
No flu:& 100 %-15 %= 85 %
Let's now add all of the obtained information to our table.
Got the Flu
Did Not Get the Flu
Totals
Vaccinated
6 %
54 %
60 %
Not Vaccinated
40 %
Totals
15 %
85 %
100 %
Now we can use the already obtained information to find the two missing relative frequencies. In the row Got the Flu the percentage of vaccinated people, 6 %, and the percentage of not vaccinated people add up to 15 %. Also in the row Did Not Get the Flu the percentage of vaccinated people, 54 %, and the percentage of not vaccinated people add up to 85 %.
Not vaccinated, flu:& 15 %-6 %= 9 %
Not vaccinated, no flu:& 85 %-54 %= 31 %
Finally, we can complete our table.
Got the Flu
Did not Get the Flu
Totals
Vaccinated
6 %
54 %
60 %
Not Vaccinated
9 %
31 %
40 %
Totals
15 %
85 %
100 %
b Now we want to find the probability of getting the flu, given that a person is vaccinated. Note that this is a conditional probability. Consider the completed table.
Got the Flu
Did Not Get the Flu
Totals
Vaccinated
6 %
54 %
60 %
Not Vaccinated
9 %
31 %
40 %
Totals
15 %
85 %
100 %
The condition of a person being vaccinated limits the relative frequency of total possible outcomes to 60 %= 0.6. The relative frequency for the fact that a person gets the flu and is vaccinated is 6 %= 0.06. To calculate the desired conditional probability we need to divide these two numbers.
P(flu|vaccinated)=0.06/0.6=0.1
⇓
P(flu|vaccinated)=10 %
The probability of getting the flu given that the person is vaccinated is 10 %.
c Finally, we want to find the probability of getting the flu given that a person has not been vaccinated. Note that this is a conditional probability. Consider the completed table.
Got the Flu
Did Not Get the Flu
Totals
Vaccinated
6 %
54 %
60 %
Not Vaccinated
9 %
31 %
40 %
Totals
15 %
85 %
100 %
The condition of a person not being vaccinated limits the relative frequency of total possible outcomes to 40 %= 0.4. The relative frequency for the fact that a person gets the flu and is not vaccinated is 9 %= 0.09. To calculate the desired conditional probability we need to divide these two numbers.
P(flu|not vaccinated)=0.09/0.4=0.225
⇓
P(flu|not vaccinated)=22.5 %
The probability of getting the flu given that the person has not been vaccinated is 22.5 %.
