MATH 1401 — Section 5.1 — Basic Concepts of Probability
MATH 1401 · Dr. Barry Monk

Basic Concepts of Probability

Section 5.1

Probability Experiment

A probability experiment is one where we cannot predict the outcome of a single trial, but we can anticipate the overall pattern after many repetitions.

When flipping a fair coin, a single toss is uncertain, but over many tosses, about half will be heads and half tails. The probability of an event is the long-term proportion of times it occurs. For a fair coin, the probability of heads or tails is 1/2.

Proportion of heads 0.4994 Repetitions 10,000

The Law of Large Numbers states that as a probability experiment is repeated many times, the proportion of occurrences of a given event will get closer to its expected probability.

Two red dice with white pips, drawn as a flat illustration, one showing five and one showing six.

Sample Space

The collection of all the possible outcomes of a probability experiment is called a sample space.

Example:

A coin is tossed

Sample space: {Heads, Tails}

A student is selected at random from a list of 10,000 at a university

Sample space: The collection of the 10,000 students

A simple random sample of 100 people is selected from a list of 10,000 subscribers to a YouTube channel

Sample space: Every group of 100 people that can possibly be chosen

Definition

Events

We often focus on events that include several outcomes. For example, when rolling a die, the probability of rolling an odd number involves the outcomes {1, 3, 5} from the sample space {1, 2, 3, 4, 5, 6}. A collection of outcomes from a sample space is called an event.

Two red dice with white pips, drawn as a flat illustration, one showing five and one showing six.

Probabilities for events are denoted by P. For instance, the probability of a coin landing heads is P(Heads)=1/2. More generally, if A is an event, its probability is written as P(A).

The probability of an event is always between 0 and 1 (inclusive).

If A cannot occur, then P(A)=0.

If A is certain to occur, then P(A)=1.

Probabilities with Equally Likely Outcomes

If a sample space has n equally likely outcomes and an event A has k outcomes, then

P(A)=Number of outcomes in ANumber of outcomes in the sample space=kn
Example: The Georgia Lottery “Cash 4” wordmark, in green and orange lettering over a fan of dollar bills.

In the Georgia Cash-4 Lottery, a winning number between 0000 and 9999 is chosen at random, with all the possible numbers being equally likely. What is the probability that all four digits are the same?

Solution:

The outcomes in the sample space are the numbers from 0000 to 9999, so there are 10,000 equally likely outcomes in the sample space.

There are 10 outcomes for which all the digits are the same: 0000, 1111, 2222, and so on up to 9999.

P(All four digits the same)=1010,000=0.001
0000 1111 2222 3333 4444 5555 6666 7777 8888 9999
Example

Probabilities with Equally Likely Outcomes

A penny, a nickel, and a dime are tossed. There are eight possible outcomes: HHH, HHT, HTH, HTT, THH, THT, TTH, and TTT. Assume these outcomes are equally likely.

What is the probability that all three tosses are the same?

The obverse of a United States penny: Lincoln in profile facing right, “IN GOD WE TRUST” above and the date 2013. The obverse of a United States nickel: Jefferson facing forward, “Liberty” and the date 2023. The obverse of a United States dime: Roosevelt in profile facing left, “LIBERTY” curved at the left and the date 2017.
H H H
H H T
H T H
H T T
T H H
T H T
T T H
T T T

Solution: Of the eight equally likely outcomes, the two outcomes HHH and TTT correspond to having all tosses the same. Therefore,

P(All three tosses are the same)=28=14
Example

Probabilities with Equally Likely Outcomes

A penny, a nickel, and a dime are tossed. There are eight possible outcomes: HHH, HHT, HTH, HTT, THH, THT, TTH, and TTT. Assume these outcomes are equally likely.

Suppose that we know that the penny lands tails. What is the probability of getting more tails than heads?

The obverse of a United States penny: Lincoln in profile facing right, “IN GOD WE TRUST” above and the date 2013. The obverse of a United States nickel: Jefferson facing forward, “Liberty” and the date 2023. The obverse of a United States dime: Roosevelt in profile facing left, “LIBERTY” curved at the left and the date 2017.
H H H
H H T
H T H
H T T
T H H
T H T
T T H
T T T

Solution: If we know that the penny lands tails, then the sample space is: THH, THT, TTH, and TTT. Three of these four outcomes correspond to “more tails than heads” so:

P(More tails than heads)=34
An empty television game-show set lit in blue and amber, tiered seating curving around a circular floor with a radial spoked emblem at its center.
The Who Wants to Be a Millionaire logo disc.

Which planet in our solar system takes longest to rotate once on its axis?

No lifelines on this one.

The four options are equally likely to a guesser, so P(Correct guess)=14.

An empty television game-show set lit in blue and amber, tiered seating curving around a circular floor with a radial spoked emblem at its center.
The Who Wants to Be a Millionaire logo disc.

Which of these lakes holds the greatest depth of any lake on Earth?

50:50 removes two wrong answers.

After the 50:50, two options remain, so P(Correct guess)=12.

An empty television game-show set lit in blue and amber, tiered seating curving around a circular floor with a radial spoked emblem at its center.
The Who Wants to Be a Millionaire logo disc.

Which of these summits sits farthest from the center of the Earth?

A phoned friend rules out one wrong answer.

A phoned friend rules out one wrong answer, so three options remain and P(Correct guess)=13.

Example: Sampling as a Probability Experiment

There are 10,000 families in a certain town categorized as follows:

Own a house Own a condo Rent a house Rent an apartment
4753 1478 912 2857

A pollster samples a single family from this population.

What is the probability that the sampled family owns a house?

The sample space consists of 10,000 households. Of these, 4753 own a house. Therefore,

P(Own a house)=475310,000=0.4753

What is the probability that the sampled family rents?

The number of families who rent is 912+2857=3769. Therefore,

P(Rents)=376910,000=0.3769
A sunlit gymnasium with dozens of basketballs scattered across the floor and two in mid-air near the hoop.
Definition

Empirical Method

The Law of Large Numbers states that as we repeat a probability experiment many times, the proportion of a particular outcome will tend to get closer to the true probability of that outcome.

The Empirical Method involves repeating an experiment multiple times and using the observed proportion of an outcome to estimate its probability.

A hospital nursery with four newborns swaddled in white in clear bassinets under warming lamps, a gloved hand adjusting one baby’s cap.

Example: Empirical Method

In a recent year, there were 313,752 births in the U.S. to low birth weight babies, while 3,477,960 were to babies with normal weight. Approximate the probability that a newborn baby is low birth weight.

Solution: We compute the number of times the experiment has been repeated:

3,477,960+313,752=3,791,712 births.

The proportion of low birth weight babies is

313,7523,791,712=0.0827

We approximate P(Low birth weight)≈0.0827.

Three white dice stacked one on top of another on a dark surface, lit from the right against a black background. The top die shows three pips on the face toward the camera and six on its right side; the middle die shows five.
Definition

Simulation

Sometimes we cannot repeat an experiment enough times to use the Empirical Method.

When that happens, we can use technology to run an equivalent experiment many times instead. Running a virtual experiment this way is called simulation.

Example: Simulation

If three dice are rolled, the smallest possible total is 3 and the largest is 18. We will simulate 1000 rolls of three dice to estimate the probability that the total is 10.

Each bar counts how often a total came up. A total of 10 came up 129 times, so we estimate P(Total = 10)=129/1000=0.129.

Roll again runs a fresh 1000 rolls. The estimate changes every time — it is a sample, not the answer.

Of the 216 equally likely outcomes, 27 give a total of 10, so the exact probability is 27/216=0.1250 — the dashed line.

Now drag the number of rolls: few rolls, a jumpy estimate; many rolls, one that settles on the line.

Rolls 1,000 Gave 10 129 Estimate of P(10) 0.129 Number of rolls
Definition

Unusual Events

An unusual event is one that is not likely to happen. In other words, an event whose probability is small. We will use the following rule of thumb.

Any event whose probability is less than 0.05 is considered to be unusual.

A single dew-beaded four-leaf clover standing out from a dense bed of ordinary three-leaf clover.
A tiered lecture hall full of seated students, with one student standing among them wearing a T-shirt printed with a large π.

Example

About 11% of the U.S. population is left-handed. You randomly select one person and they are left-handed. Is this unusual?

Because 0.11 is greater than 0.05, selecting a left-handed person is not unusual.

In a college of 5000 students, 150 are math majors. A student is selected at random and turns out to be a math major. Is this unusual?

The event of choosing a math major consists of 150 students out of a total of 5000. The probability of choosing a math major is

1505000=0.03

This would be considered an unusual event.

A small silver two-door city car parked at the curb between two larger cars, in front of a red-brick building.

The Rule of Complements

Rule of Complements
P(Ac)=1−P(A)
Example:

According to the Wall Street Journal, 40% of cars sold in a recent year were small cars. What is the probability that a randomly chosen car sold in that year is not a small car?

P(Not a small car)=1−P(Small car)=1−0.40=0.60
Check Your Understanding

Example

Two hundred students were enrolled in a Statistics class.

Exactly 50 of them are sophomores.

The complement is that the number of sophomores is not 50.

A movie theater has 150 seats.

Exactly 100 seats are occupied.

The complement is that the number of occupied seats is not 100.

Check Your Understanding

Example

A batch contains 500 manufactured parts.

At least 10 parts are defective.

The complement is that fewer than 10 parts are defective.

A parking garage has 300 spaces.

More than 250 spaces are filled.

The complement is that 250 or fewer spaces are filled.

Check Your Understanding

Example

A Spotify playlist has 50 songs.

Fewer than 20 have been played.

The complement is 20 or more songs have been played.

A warehouse stores inventory in 200 bins.

At most, 25 bins are empty.

The complement is more than 25 bins are empty.

The view from a rideshare passenger seat: the driver in a cap and red hoodie with one hand on the wheel, a phone in a dashboard mount showing a navigation map, and freeway overpasses ahead through the windscreen.
Example

Naming a Complement, Then Finding Its Probability

A rideshare company records 300 rides on a Friday night. The probability that 50 or more of them are surge-priced is 0.35.

What is the probability that at least 50 rides are surge-priced?

“At least 50” is another way of saying “50 or more”, so this is given:

P(At least 50)=0.35

What is the probability that fewer than 50 rides are surge-priced?

“Fewer than 50” is the complement of “50 or more”, so

P(Fewer than 50)=1−0.35=0.65
You Are Ready For:

5.1 HW: Basic Concepts of Probability

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