Download Venn Diagrams (7.2)

7.2 Probability using Two-Way Tables and Venn Diagrams
Key Terms:
Event:
Sample Space:
Basic Rules of Probability:
1) The probability of any event occurring is between ________________.
2) All possible outcomes together must have a probability equal to ______.
3) The probability that an event does not occur is equal to __________________.
4) If two events, A and B, have no outcomes in common, the probability of either
event (A or B) occurring is ___________________.
5) If all outcomes are equally likely, the probability of an event occurring is
P(event) =
_______________________
Mutually Exclusive:
General Rule:
If A and B are any two events resulting from some chance process,
then,
P(A and B) = P (A ∩ B) _____________________________
P(A or B) = P (A ∪ B) ______________________________
Not Mutually Exclusive:
General Rule:
If A and B are any two events resulting from some chance process,
then,
P(A or B) = P (A ∪ B) ______________________________
Probabilities written symbolically:
P(A) =
P(B) =
P( Ac ) =
𝑃(𝐵! ) =
P( A ∪ B) =
𝑃(𝐴 ∩ 𝐵) =
P( A ∩ B c ) =
𝑃(𝐴 ∪ 𝐵! ) =
Two-Way Table:
Venn Diagram:
Example
Who washes their hands after using the bathroom?
Mr. Starnes surveyed the students in his introductory statistics class at The College of New
Jersey about their hand washing habits. The two-way table summarizes the students’
responses.
Percent of time that student washes hands after using
bathroom
Sex
Female
Male
25% - 49%
0
1
50% - 74%
2
0
75% - 99%
6
5
100%
13
1
Suppose Mr. Starnes chooses a student at random from the class. Find the probability that:
(a) the student is male.
(b) the student did not report always washing his or her hands after using the bathroom.
(c) the student is female and reported washing her hands after 75% to 99% of bathroom
visits.
(d) the student reported washing his or her hands after 75% to 99% of bathroom visits or is
male.
Exercise Choosing at random Abby, Betty, Cara, Dan, and Evan work in a firm’s public
relations office. Their employer must choose two of them to attend a conference in Paris. To
avoid unfairness, the choice will be made by drawing two names from a hat.
(a) Write down all possible choices of two of the five names. These are the possible
outcomes.
(b) The random drawing makes all outcomes equally likely. What is the probability of each
outcome?
(c) What is the probability that Cara is chosen?
(d) What is the probability that neither of the two men (Dan and Evan) is chosen?
Exercise Rolling two dice Refer to the sample space for all outcomes of rolling two dice
above. Let Event A = number of outcomes where at least one of the die is a 3 and let Event
B = the number of outcomes that have a sum of at least five.
a) Fill in the two-way table below
Sum is
at least a 5
Sum is
less than 5
Total
3 on at least one Die
3 on no dice
Total
b) Fill in the Venn Diagram of the same situation at the right
c) Find the following probabilities:
P(A) = ________
P(B) = ________
P(A c ) = _________
P(B c ) = ________
P(A ∪ B) = _________
P(A ∩ B) = __________
P(A c ∪ B) = _________
P(A c ∩ B) = __________
P(A ∪ B c ) = _________
P(A ∩ B c ) = __________
P(A c ∪ B c ) = _________
P(A c ∩ B c ) = _________
A
B