Download 5.1 Probability overview (Answer in notes)

5.1 PROBABILITY OVERVIEW (PUT IN NOTES)
1. A laundry bag contains 5 red, 9 blue, and 6 white socks. Find each probability below:
a. 𝑃(𝑟𝑒𝑑)
b. 𝑃(𝑛𝑜𝑡 𝑟𝑒𝑑)
2. What is the probability that a given month of the year begins with the letter “J”?
3. Two dice are tossed. What is the probability of both dice being a factor of 12?
4. Two dice are tossed. What is the probability the sum being greater than 9?
PROBABILITY RULES
5.2
SAMPLE SPACE AND PROBABILITY MODELS
 The sample space S of a chance process is the set of all possible outcomes.
 A probability model is a description of some chance process that consists of two parts:
•
a sample space S and
•
a probability for each outcome
•
Total probability adds to 1
 Examples: For each define the sample space and show this is a legitimate probability model.
 One Die: S =
 Flipping a coin 3 times: S =
BASIC RULES OF PROBABILITY
• The probability of any event is a number between 0 and 1.
• All possible outcomes together must have probabilities whose sum is exactly 1.
• If all outcomes in the sample space are equally likely, the probability that event A occurs can be found
using the formula
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑓𝑜𝑟 𝑒𝑣𝑒𝑛𝑡 𝐴
𝑃 𝐴 =
𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒
• The probability that an event does not occur is 1 minus the probability that the event does occur.
•
Complement:
• If two events have no outcomes in common, the probability that one or the other occurs is the sum of
their individual probabilities. These events are said to be mutually exclusive or disjoint.
•
Addition rule for mutually exclusive events:
VENN DIAGRAMS AND PROBABILITY
Because Venn diagrams have uses in other branches of mathematics, some standard vocabulary and
notation have been developed.
The complement AC contains
exactly the outcomes that
are not in A.
The events A and B are mutually exclusive
(disjoint) because they do not overlap.That
is, they have no outcomes in common.
VENN DIAGRAMS AND PROBABILITY
The intersection of events A and B (A ∩ B) is the
set of all outcomes in both events A and B.
The union of events A and B (A ∪ B) is the set of
all outcomes in either event A or B.
Hint:To keep the symbols straight, remember ∪ for union and ∩ for intersection.
TWO-WAY TABLE AND VENN DIAGRAM EXAMPLE
Students in a college statistics class wanted to find out how common it is for young adults to have their ears
pierced. They recorded data on two variables – gender and whether the student had a pierced ear – for all
178 people in the class, 90 males and 88 females.
 Create a two-way table, given that 19 males and 84 females have their ears pierced.
 Create a Venn Diagram where event A is males and event B is pierced ears.
TWO-WAY TABLE AND VENN DIAGRAM EXAMPLE CONT.
Remember 𝐴 = 𝑚𝑎𝑙𝑒𝑠 and 𝐵 = 𝑝𝑖𝑒𝑟𝑐𝑒𝑑 𝑒𝑎𝑟𝑠. Find each of the following probabilities:
1. 𝑃 𝐴 ∩ 𝐵 =
2. 𝑃 𝐴 ∩ 𝐵𝐶 =
3. 𝑃 𝐴𝐶 ∩ 𝐵 =
4. 𝑃 𝐴𝐶 ∩ 𝐵𝐶 =
PIERCED EARS CONTINUED….GENERAL ADDITION RULE
 Are events A and B mutually exclusive/disjoint?
If A and B are any two events resulting from some chance
process, then
P(A or B) = P(A) + P(B) – P(A and B)
So 𝑃 𝐴 ∪ 𝐵 =