2.1 Basic Probability and the Addition Rule
Fraction Review
- Simplify the fractions.
- \(\frac{12}{40}\)
\(\frac{3}{10}\)
- \(3 \frac{5}{15}\)
\(3 \frac13\)
- \(\frac{72}{48}\)
\(\frac32\)
- \(\frac{12}{40}\)
- Evaluate each of the following.
- \(\frac{3}{4} \times \frac{2}{7}\)
\(\frac3{14}\)
- \(\frac{18}{25} \times \frac{17}{24}\times \frac{16}{23}\)
\(\frac{204}{575}\)
- \(\frac{3}{4} \times \frac{2}{7}\)
- Voting by Age Group in the 2020 U.S. Presidential Election
Age Group
Voted (in thousands)
Did Not Vote (in thousands)
Total (in thousands)
18-34
37,868
35,662
73,530
45-64
53,653
28,260
81,913
65+
39,742
15,532
55,274
Total (in thousands)
131,263
79,454
210,717One person is randomly selected from the observational study of voters in the 2020 U.S. Presidential election. Using the data in the table, what is the probability that the person chosen:
- is 18-34?
\(\frac{73,530}{210,717}= 0.349\) - is 65+?
\(\frac{55,274}{210,717}= 0.2622\) - is a 45-64-year-old voter?
\(\frac{53,653}{210,717}= 0.2546\) - is an 18-34-year-old non-voter?
\(\frac{35,662}{210,717}= 0.1692\) - did not vote?
\(\frac{79,454}{210,717}= 0.3771\) - did vote?
\(\frac{131,263}{210,717}= 0.6229\) or \(1-0.3771=0.6229\)
The Addition Rule: Probability of Event A OR Event B occurring.
Complete this statement for the addition rule: P(A or B) =
\(P(A \; \text {or} \; B)=P(A)\;+\;P(B)\;-\;P(A \; \text {and} \; B)\)
\(P(A\; \text {or} \;B)=P(A)\;+\;P(B)\) if A and B are mutually exclusive
Using the Addition Rule with a Contingency Table
- The following table gives the number of people with health insurance coverage in the U.S. in 2022. Note that the numbers are in millions of people. Source of data
(In millions) Private insurance Public Insurance Uninsured Total Number Children age 0-17 44.6 27.5 4.1 76.2
Adults age 18-64 137.3 37.4 21.2 195.9
Adults age 65 and older 3.1 54.2 0.6 57.9
Total Number 185
119.1
25.9
330
Find the probabilities if you randomly select one person:
- P(Child) =
\(\frac{n(E)}{n(S)}=\frac{76.2}{330}=0.2309\)
- P(Public Insurance) =
\(\frac{119.1}{330}=0.3609\)
- P(Child with Public Insurance) =
\(\frac{27.5}{330}=.0833\)
- P(Child or Public Insurance) =
\(P(\text {child})+P(\text {public})-P(\text { child AND public})=\frac{76.2}{330}+\frac{119.1}{330}-\frac{27.5}{330}=\frac{167.8}{330}=0.5085\)
- P(Public Insurance or Private Insurance) =
These events are mutually exclusive. \(\frac{119.1}{330}+\frac{185}{330}=\frac{304.1}{330}=0.9215\)
- P(Adult Age 18-64 or Private Insurance) =
\(\frac{195.9}{330}+\frac{185}{330}-\frac{137.3}{330}=\frac{243.6}{330}=0.7382\)
- P(Uninsured or not a Child)
\(P(\text {uninsured}) + P(\text {not a child}) - P(\text {uninsured AND not a child}) =\frac{25.9}{330}+\frac{195.9}{330}+\frac{57.9}{330}-\frac{21.2}{330}-\frac{0.6}{330}=\frac{257.9}{330}=0.7815\)
- is 18-34?
Calculating the Probability of an Event
How do you calculate the probability of an event? \(P(E)\)=__________
What is the probability of an impossible event? __________
What is the probability of a sure event? _________
What does this statement mean? \(0 \leq P(E) \leq 1\)
The complement of an event is the “opposite” of that event. If the event says that something will occur, then the complement of the event is that the thing will not occur.
What is the complement of the event “it will rain today”?
What is the complement of the event “earned an A in the class”?
What is the complement of the event “at least one student was late for class”?
If an event is denoted by E, the complement is represented by any of the notations \(E',\;\overset-{E,}\;or\;E^c\),
Complete this statement: \(P(E)+P(\overset-{E)}\)= __________