Probability: Sample Space, Outcomes, and Events

Probability:

Sample Space, Outcomes, and Events

Objectives

1. Define probability, statistical experiment, sample space, outcomes, and events

2. Find probabilities of events

3. Differentiate each type of probability

​21 (2008) - Counting Cards Scene

Blaise Pascal

Pierre de Fermat

Statistical Experiment

Characteristics of a Statistical Experiment

1. The experiment can have more than one possible outcome.

2. Each possible outcome can be specified in advance.

3. The outcome of the experiment depends on chance.

Multiple Choice

Is tossing a fair coin a statistical experiment?

• Yes

• No

Outcome

The result of a single attempt in a statistical experiment is called an outcome.

Open Ended

A coin is tossed once. What are the possible outcomes?

Sample Space (S)

The set of all possible outcomes in a statistical experiment is called a sample space.

Open Ended

Determine the sample space of tossing a coin twice.

Event (E)

An event is a set of outcomes in a statistical experiment - essentially a subset of the sample space.

Probability of an Event

It is the likelihood of an event from happening.

Classical Probability

The probability is computed even before an actual experiment is done, because the possible outcomes in the sample space are known and are assumed to be equally likely.

​-> Number of outcomes in E

-> Number of outcomes in S

​Probability of E ->

Open Ended

What is the probability of selecting an ace in a well-shuffled deck of cards?

Empirical Probability

Unlike the classical probability, empirical probability depends on performing the actual statistical experiment, wherein we need to record the results. This type of probability uses frequencies to find the probability of an event and it is based on observation.

​-> frequency of the class

-> sum of the frequencies

Open Ended

A medical practitioner in a certain city recorded the number of CoViD-19 patients as shown in the table. What is the probability of selecting a CoViD-19 patient who is 35-49 years old?

Subjective Probability

Example: A meteorologist estimates that there is an 80% chance of rain at 2pm tomorrow.

This type of probability is based on past experiences or intuition of a person or group.

Properties of Probability

1. The probability of an event P(E) if its occurrence is certain is 1, that is 𝑷(𝑬) = 𝟏.

2. The probability of an event P(E) if its occurrence is impossible is 0, that is 𝑷(𝑬) = 𝟎.

3. 0 ≤𝑷(𝑬) ≤ 1.

4. The sum of the probabilities of all outcomes in the sample space is 1.

Example: Roll a fair die once.

Multiple Choice

A box contains rubber bands of which 20 are blue, 30 are red, 45 are green, and 15 are yellow. What is the probability of selecting a green rubber band?

• $$\frac{9}{22}$$

• $$\frac{2}{11}$$

• $$\frac{3}{22}$$

• $$\frac{3}{11}$$

Multiple Choice

A box contains rubber bands of which 20 are blue, 30 are red, 45 are green, and 15 are yellow. What is the probability of selecting a red or blue rubber band?

• $$\frac{2}{11}$$

• $$\frac{3}{11}$$

• $$\frac{4}{11}$$

• $$\frac{5}{11}$$

Multiple Choice

The distribution of the scholarship of students in a certain school is shown below. What is the probability that a randomly chosen student is an academic scholar?

• $$\frac{4}{29}$$

• $$\frac{6}{29}$$

• $$\frac{49}{145}$$

• $$\frac{46}{145}$$

Multiple Choice

A card is chosen from a deck. What is the probability of selecting a black face card?

• $$\frac{3}{52}$$

• $$\frac{3}{26}$$

• $$\frac{3}{13}$$

• $$\frac{1}{13}$$

Summary

Probability is the chance of an event occurring which is a value from 0 to 1. Probabilities can be interpreted as classical, empirical or subjective.

References