7A Coreq

In the next preview assignment and in the next class, you will need to calculate the probability of an event occurring and conduct simulations to estimate probabilities.

Basic Probability Rules

A chance experiment involves making an observation in a situation where there is uncertainty about which of two or more possible outcomes will result.

Question 1

Suppose we flip a fair coin. What are the possible outcomes of the coin flip (assuming that it cannot land on an edge)?

This list of all possible outcomes of a chance experiment is called the sample space.
In some situations, all of the possible outcomes of a chance experiment occur with the same probability. For example, when a fair six-sided die is rolled, the numbers 1, 2, 3, 4, 5, and 6 are all equally likely to occur (we say these are “equally likely outcomes”). When dealing with equally likely outcomes, it is sometimes helpful to list (or count) all of the possible outcomes.
For a chance experiment, we are often interested in how likely a particular outcome (or collection of outcomes) is. An outcome or collection of outcomes for a chance experiment is called an event. The probability of an event is a numeric measure of how likely the event is to happen. Note the conventional notation P(event) indicates the probability of an event.
When the outcomes are equally likely, we can use the following formula to calculate the theoretical probability of event A:

Notice that a probability can be determined by thinking of it as two counting problems followed by the computation of a related fraction.

probability of A = P(A) = number of outcomes in event A number of all possible outcomes

Question 2

What is the probability of the coin landing on “heads?” Write your answer as a fraction, a decimal, and a percentage.

A probability is always a number from 0 to 1, inclusive (which means that 0 and 1 are included). Probability may be written as a percentage, from 0% to 100%, inclusive. We can also express a probability as a fraction or a decimal.

Question 3

What is the probability of the coin landing on “heads?” Write your answer as a fraction, a decimal, and a percentage.

Sometimes we deal with situations that are more complicated than rolling a die or flipping a coin. However, the idea is the same—we think of the probability problem as two counting problems followed by the computation of a related fraction.

Question 4

A shipping company delivered 25,000 packages last week, and 24,500 of the packages arrived on time.

  1. If you had been expecting one of those packages, what is the probability that your package did arrive on time? Write your answer as a fraction, a decimal,
    and a percentage.
  2. If you had been expecting one of those packages, what is the probability that your package did NOT arrive on time? Write your answer as a fraction, a decimal,
    and a percentage.

Question 5

Suppose you guess on a True/False question on a quiz. What is the probability of guessing correctly? Write your answer as a fraction, a decimal, and a percentage.

Question 6

Suppose there are 10 True/False questions on the quiz and you guess on all 10 questions. How many questions do you anticipate answering correctly?

Question 7

Suppose there are 30 True/False questions on the quiz and you guess on all 30 questions. How many questions do you anticipate answering correctly?

Sometimes we want to estimate the probability of an event, so we simulate a chance experiment to look at how often the event occurred and estimate the probability.

Question 8

Suppose we want to investigate the number of questions a student would answer correctly on a 10-question True/False quiz. Let’s conduct a chance experiment to see how many questions a student would get correct.

  1. Describe how you would conduct the chance experiment with a fair coin.
  2. Describe how you would conduct the chance experiment with a fair six-sided die.
  3. Describe how you would conduct the chance experiment with a random number generator that selected a whole number from 0–9.

Question 9

Conduct one of the chance experiments from Question 8 using the DCMP Random Number Generator.

Direct link: https://dcmathpathways.shinyapps.io/RandomNumbers/

Use the following inputs:

  • Choose Minimum: Depends on experiment selected
  • Choose Maximum: Depends on experiment selected
  • How many numbers would you like to generate? 10
  • Sample with Replacement? Yes (because we want to look at the number and put it back, so we are always picking from all the total possible numbers)
  • Select “Frequency Table”
  • Select “Generate”

Question 10

Answer the following questions for discussion:

  • What inputs (min/max) did you select and why?
  • How many questions did the student answer correctly in your chance experiment? How did you determine that number?
  • What was the probability of answering a question correctly in your chance experiment? How did you determine that number?