1. Essential Statistics 2E
William Navidi and Barry Monk

2. Basic Concepts in Probability
Section 4.1

3. Objectives Construct sample spaces Compute and interpret probabilities
Approximate probabilities using the Empirical Method

4. Objective 1
Construct sample spaces

5. Probability
A probability experiment is one in which we do not know what any individual outcome will be, but we do know how a long series of repetitions will come out. For example, if we toss a fair coin, we do not know what the outcome of a single toss will be, but we do know what the outcome of a long series of tosses will be – about half “heads” and half “tails”. The probability of an event is the proportion of times that the event occurs in the long run. So, for a “fair” coin, that is, one that is equally likely to come up heads as tails, the probability of heads is 1/2 and the probability of tails is 1/2.

6. Law of Large Numbers
The Law of Large Numbers says that as a probability experiment is repeated again and again, the proportion of times that a given event occurs will approach its probability.

7. Sample Space
The collection of all the possible outcomes of a probability experiment is called a sample space.
Example:
Describe the sample space for each of the following:
The toss of a coin
The sample space is {Heads, Tails}.
The roll of a die
The sample space is {1, 2, 3, 4, 5, 6}
Selecting a student at random from a list of 10,000 at a large university
The sample space consists of the 10,000 students

8. Probability Model
We are often concerned with occurrences that consist of several outcomes. For example, when rolling a die, we might be interested in the probability of rolling an odd number. Rolling an odd number corresponds to the collection of outcomes {1, 3, 5} from the sample space {1, 2, 3, 4, 5, 6}. In general, a collection of outcomes of a sample space is called an event. Once we have a sample space, we need to specify a probability of each event. This is done with a probability model. We use the letter “𝑃” to denote probabilities. For example, we denote the probability that a tossed coin lands heads by 𝑃(Heads). In general, if A denotes an event, the probability of event A is denoted by 𝑃(A).

9. Objective 2
Compute and interpret probabilities

10. Probability Rules
The probability of an event is always between 0 and 1. In other words, for any event A, 0 ≤ 𝑃(A) ≤ 1.
If A cannot occur, then 𝑃(A) = 0.
If A is certain to occur, then 𝑃(A) = 1.

11. 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 A Number of outcomes in the sample space = 𝒌 𝒏 Example: 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? 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 𝑃(All four digits the same) = 10 10,000 =0.001

12. Sampling is a Probability Experiment
Sampling an individual from a population is a probability experiment. The population is the sample space and members of the population are equally likely outcomes.

13. Example: Sampling as a Probability Experiment
There 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?
What is the probability that the sampled family rents?
Solution:
The sample space consists of 10,000 households. Of these, own a house. Therefore, 𝑃(Own a house) = ,000 =
The number of families who rent is = Therefore, 𝑃(Rents) = ,000 =

14. Unusual Events
An unusual event is one that is not likely to happen. In other words, an event whose probability is small. There are no hard-and-fast rules as to just how small a probability needs to be before an event is considered unusual, but we will use the following rule of thumb. Any event whose probability is less than 0.05 is considered to be unusual. Example: 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? Solution: The event of choosing a math major consists of 150 students out of a total of The probability of choosing a math major is =0.03. This would be considered an unusual event.

15. Objective 3
Approximate probabilities using the Empirical Method

16. Approximating Probabilities – Empirical Method
The Law of Large Numbers says that if we repeat a probability experiment a large number of times, then the proportion of times that a particular outcome occurs is likely to be close to the true probability of the outcome. The Empirical Method consists of repeating an experiment a large number of times, and using the proportion of times an outcome occurs to approximate the probability of the outcome.

17. Example: Empirical Method
In a recent year, there were 2,046,935 boys and 1,952,451 girls born in the U.S. Approximate the probability that a newborn baby is a boy. Solution: We compute the number of times the experiment has been repeated: 2,046,935+1,952,451=3,999,386 births. The proportion of births that are boys is 2,046,935 3,999,386 = We approximate 𝑃(Boy) ≈

18. You Should Know . . . The Law of Large Numbers The definition of:
Probability
Sample space
Event
How to compute probabilities with equally likely outcomes
The rule-of-thumb for determining when an event A is unusual (if 𝑃(A) < 0.05)
How to approximate probabilities using the Empirical Method