Mod 3
3.1 Experiments
Experiment - a process or activity that generates observable and well-defined outcomes. Each outcome
of an experiment is called a sample point. The set of all possible outcomes an experiment can produce is
called a sample space.
Probability is a measure of the likelihood that a certain event will occur. If an event is sure to occur, its
probability is 1. If an event cannot possibly happen, its probability is 0. The probability of any event is
between 0 and 1, inclusive.
Probability rules apply to any experiment:
Rule #1: The probability for each assigned outcome must be between 0 and 1, inclusive. Let a single
event, out of i total events, be designated by Ei. The probability of an event Ei occurring is designated by
P(Ei ).
0 ≤ P(Ei ) ≤ 1 for each i
Rule #2: The sum of the probabilities for all the outcomes must equal 1. That is, if there are n total
events, then:
P(E1 ) + P(E2 ) + P(E3 ) + ⋯ + P(En ) = 1
How do we calculate probabilities?
Two methods for calculating or assigning probabilities:
1. Classical method = “equally likely to occur”
2. Relative frequency method (empirical method)
Relative Frequency Method
The relative frequency method or empirical method of assigning probabilities is used whenever
actual data is available. The frequency is calculated by dividing the number of times an event
occurred divided by the total number of trials in the experiment. That is, the probability that
event E will occur is:
,P(E) – Probability of Event
N(E) – Number of outcomes on the event E.
N(S) – Number of outcomes in the Sample space.
Example 3.2:
A certain church has 270 members. 125 of the members are from the city, 60 are from the suburbs, and
85 of the members are from the country. If you choose a member randomly from this church, what is
the probability that the member is from the city, the suburbs, and the country?
Example 3.3
A fast-food restaurant has kept track of its customers’ orders over the past month. During that period,
the fast food restaurant had 1558 customers and:
340 customers ordered only food.
655 customers ordered only drinks (coffee, soft drinks, milkshakes)
563 customers ordered food and drinks.
1) Based on this data, what is the probability that a customer who enters the restaurant will order
only food?
2) Make a probability distribution for this data.
In a certain high school, 54% of the students are female, and 46% of the students are male. If you
randomly select a student from this school, what is the probability that the student will be female?
What is the probability that the student will be male?
3)
4)
A bag contains jelly beans of various colors. In the bag, 26% of the jelly beans are red, 20% are black,
35% are green, and 19% are yellow. If you choose a jellybean randomly from the bag, what is the
probability that the jelly bean will be black? What is the probability that it will be yellow?
, 5)
Counting Techniques
Example 3.4. Suppose that you flip a coin three times. Calculate the number of possible outcomes.
Example 3.5
Consider the following three-step experiment:
Step #1: Toss a fair coin once.
Step #2: Select a card from a pile consisting of the Jack of Hearts, Queen of Hearts, King of Hearts, and
Ace of Hearts.
Step #3: Roll a fair die.
What are the total number of outcomes for this experiment?
Example 3.6.
Consider the following three-step experiment:
Step #1: Toss a fair coin.
Step #2: Select a card from a deck of 52 cards.
Step #3: Roll a fair die.
What are the total number of outcomes for this experiment?
Example 3.7.
3.1 Experiments
Experiment - a process or activity that generates observable and well-defined outcomes. Each outcome
of an experiment is called a sample point. The set of all possible outcomes an experiment can produce is
called a sample space.
Probability is a measure of the likelihood that a certain event will occur. If an event is sure to occur, its
probability is 1. If an event cannot possibly happen, its probability is 0. The probability of any event is
between 0 and 1, inclusive.
Probability rules apply to any experiment:
Rule #1: The probability for each assigned outcome must be between 0 and 1, inclusive. Let a single
event, out of i total events, be designated by Ei. The probability of an event Ei occurring is designated by
P(Ei ).
0 ≤ P(Ei ) ≤ 1 for each i
Rule #2: The sum of the probabilities for all the outcomes must equal 1. That is, if there are n total
events, then:
P(E1 ) + P(E2 ) + P(E3 ) + ⋯ + P(En ) = 1
How do we calculate probabilities?
Two methods for calculating or assigning probabilities:
1. Classical method = “equally likely to occur”
2. Relative frequency method (empirical method)
Relative Frequency Method
The relative frequency method or empirical method of assigning probabilities is used whenever
actual data is available. The frequency is calculated by dividing the number of times an event
occurred divided by the total number of trials in the experiment. That is, the probability that
event E will occur is:
,P(E) – Probability of Event
N(E) – Number of outcomes on the event E.
N(S) – Number of outcomes in the Sample space.
Example 3.2:
A certain church has 270 members. 125 of the members are from the city, 60 are from the suburbs, and
85 of the members are from the country. If you choose a member randomly from this church, what is
the probability that the member is from the city, the suburbs, and the country?
Example 3.3
A fast-food restaurant has kept track of its customers’ orders over the past month. During that period,
the fast food restaurant had 1558 customers and:
340 customers ordered only food.
655 customers ordered only drinks (coffee, soft drinks, milkshakes)
563 customers ordered food and drinks.
1) Based on this data, what is the probability that a customer who enters the restaurant will order
only food?
2) Make a probability distribution for this data.
In a certain high school, 54% of the students are female, and 46% of the students are male. If you
randomly select a student from this school, what is the probability that the student will be female?
What is the probability that the student will be male?
3)
4)
A bag contains jelly beans of various colors. In the bag, 26% of the jelly beans are red, 20% are black,
35% are green, and 19% are yellow. If you choose a jellybean randomly from the bag, what is the
probability that the jelly bean will be black? What is the probability that it will be yellow?
, 5)
Counting Techniques
Example 3.4. Suppose that you flip a coin three times. Calculate the number of possible outcomes.
Example 3.5
Consider the following three-step experiment:
Step #1: Toss a fair coin once.
Step #2: Select a card from a pile consisting of the Jack of Hearts, Queen of Hearts, King of Hearts, and
Ace of Hearts.
Step #3: Roll a fair die.
What are the total number of outcomes for this experiment?
Example 3.6.
Consider the following three-step experiment:
Step #1: Toss a fair coin.
Step #2: Select a card from a deck of 52 cards.
Step #3: Roll a fair die.
What are the total number of outcomes for this experiment?
Example 3.7.
