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5. Probability
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United States of AmericaVirginia
Grade 8

5.01 Introduction to independent and dependent events

Lesson
Practice

Independent and dependent events

The probability of an event is a measure of the likelihood that the event occurs. So far, we have only considered the probability of a single event occurring. Now, we will now study the probability of two events occurring. Two events can either be independent of each other or dependent on each other.

Independent events

The outcome of one event does not influence the occurrence of the other event

When two events are independent, the chances of one event happening are not changed by the outcome of the other event. For example, when we flip a coin, there is always a \dfrac{1}{2} chance that it will land on heads. It doesn't matter whether you tossed a head or tail on the first flip.

Other examples are two independent events are:

  • Rolling two dice: the outcome of the first die does not affect the outcome of the second die

  • Spinning a spinner and flipping a coin: the outcome of the spinner does not affect the outcome of flipping the coin

  • Choosing a marble from a bag, replacing the marble, and selecting again: the selection of the first marble has no affect on the selection of the second marble

Dependent events

The outcome of one event has an impact on the outcome of the other event

When two events are dependent, the chance of one event changes depending on the outcome of the other event. For example, let's say you have an electronic claw machine game full of all different prizes. If you pull a toy car out, could someone else choose that same prize?

Other examples of two dependent events are:

  • Choosing a card, and selecting again without replacing the first card: the selection of the second card will depend on which card was drawn first

  • Taking two flights to reach a destination: making it to the second flight on time will depend on whether or not the first flight was delayed

Sample space

All possible outcomes of an experiment

We can determine all the possible outcomes of an experiment with two events using a list, chart (array), or tree diagram.

Examples

Example 1

Determine whether the selections in each experiment are independent or dependent.

a

A teacher has a "prize bag" filled with different prizes. The students form a line to draw a prize from the bag at random. Once a student has drawn a prize, they take it back to their desk.

Worked Solution
Create a strategy

Determine if the chances for an event changes as a result of the previous event.

Apply the idea

Since each student's prize is kept and not returned to the bag, it will affect the chance of selecting a particular item from the prize bag. As a result, the events are dependent.

b

A card is randomly selected from a normal deck of cards, and then returned to the deck. The deck is shuffled and another card is selected.

Worked Solution
Create a strategy

Determine if the chances for an event changes as a result of the previous event. In this case, will the card selected second be affected by the first card selected?

Apply the idea

Each card drawn is returned back into the deck, so the chances of picking a certain card on the following draw are unaffected. The events are independent.

Example 2

This spinner is spun and a six-sided die is rolled.

A 6-sided die and a spinner with 4 sections labeled 2,3,5, and 8.
a

Create an array that shows the sample space for the experiment.

Worked Solution
Create a strategy

Place the outcomes of the die on the left side of the array and the outcomes for the spinner along the top of the array. Then, fill each cell with the outcomes of each.

Apply the idea

The array for the result of spinning the spinner and rolling a six-sided die is given by:

8235
11,81,21,31,5
22,82,22,32,5
33,83,23,33,5
44,84,24,34,5
55,85,25,35,5
66,86,26,36,5
Reflect and check

Recall that the number of outcomes can be determined using the fundamental (basic) counting principle. This principle tells us that the total number of outcomes in the sample space is the product of the outcomes for each event.

There are 6 possible outcomes of the die, and 4 possible outcomes of the spinner, so there are {6\cdot 4=24} total possible outcomes.

b

Let A be the event that the die is rolled.

Let B be the event that the spinner is spun.

Are the events A and B independent or dependent?

Worked Solution
Create a strategy

Consider whether the outcome from rolling the die will affect the outcome of spinning the spinner.

Apply the idea

The number rolled on the die will have no effect on which number is spun on the spinner.

Events A and B are independent.

Idea summary

To determine whether two events are independent or dependent:

  • If the events are affected by what has already happened, they are dependent upon each other.

  • If a previous event makes no difference to what can happen in the future, they are independent of each other.

With and without replacement

When determining whether two events that occur in sequence (one after the other) are independent or dependent, we can look at what happens to the sample space for each event.

Exploration

The jar of marbles shown contains 7 red marbles, 3 blue marbles, and 10 green marbles.

A jar containing different colors of marbles. 7 marbles are red, 3 are blue, and 10 are green.

  1. A red marble is drawn from the jar, then placed back into the jar. How many marbles are in the jar?

  2. A green marble is removed from the jar. How many marbles are left in the jar?

  3. Suppose one marble is drawn and placed back into the jar, then a second marble is drawn. Let event A represent drawing a red marble first and event B represent drawing a green marble in the second draw. Are these events independent or dependent? Explain.

  4. Suppose one marble is drawn, then a second marble is drawn. The first marble was not replaced. Are events A and B independent or dependent? Explain.

Selecting items with replacement means putting each selected item back before you select another item. When we select with replacement, the number of possible outcomes at each stage stays the same.

For example, suppose we want to select two pens from a pencil case that has two blue pens and one red pen.

A tree diagram showing the probability of selecting two pens from a set of two blue pens and one red pen. A pen is replaced before selecting the 2nd pen. Ask your teacher for more information.

For the choice of the first pen, there are 3 possible outcomes: a blue pen, a blue pen, and a red pen.

If we select a pen and replace it before we make the second selection, the are still 3 pens that we could choose from.

Because the first pen was replaced, the possible outcomes for the first selection and the possible outcomes for the second selection are the same, as shown in the tree diagram.

Selecting items without replacement means that we do not put each selected item back before selecting another. When we select without replacement, the number of possible outcomes at each stage will change.

A tree diagram showing the probability of selecting two pens from a set of two blue pens and one red pen. Ask your teacher for more information.

In the same pencil case, if we select a pen and do not replace it, then there are only 2 pens remaining for our second selection.

This means that the number of possible outcomes changed from 3 pens for the first selection to 2 pens for the second selection.

In other words, the number of possible outcomes for the selection of the second pen will depend on what color was chosen first.

When constructing tree diagrams for experiments without replacement, we must think carefully about the number of each item left after each selection, and change our sample space accordingly.

A tree diagram showing the probability of selecting two pens from a set of two blue pens and one red pen. Ask your teacher for more information.
With replacement
A tree diagram showing the probability of selecting two pens from a set of two blue pens and one red pen. Ask your teacher for more information.
Without replacement

In general, events that occur with replacement are independent of each other. Events that occur without replacement are dependent upon the outcome of the first event.

Examples

Example 3

The tree diagram shows all the ways a captain and a co-captain can be selected from Matt, Rebecca and Helen.

A directed graph with three vertices labeled as Matt, Rebecca, and Helen. The vertices are connected by directed edges, indicating relationships or connections from one vertex to another. Each name appears multiple times, connected in various configurations to indicate possibly different types of interactions or flows from one individual to another. The arrows between the vertices show the direction of the relationship or connection.

Are the events of selecting a captain and a co-captain independent or dependent?

Worked Solution
Create a strategy

Any of the three people can be chosen as captain. Once a captain has been selected, there are only two remaining people that can be selected as co-captain.

Apply the idea

The number of possible outcomes between the first selection and the second selection are different. The are 3 people that could be chosen as captain, but only 2 people can be chosen from for co-captain. This means the second selection is affected by the first selection.

Since the person that gets selected as co-captain will depend on who is selected as captain, the events are dependent.

Example 4

Four cards numbered 1 to 4 are placed face down on a table. Two cards are drawn with replacement.

a

Construct a tree diagram of this situation.

Worked Solution
Create a strategy

Since the two events occur with replacement, the number of options for both draws will be the same.

Apply the idea

For the first selection, we have 1,\, 2,\, 3,\, 4 as possible options.

Since the first card is replaced, the options will stay the same for the second selection.

Tree diagram with first selection numbers 1 2 3 4 and second selection 1 2 3 4 for each first selection. Ask your teacher for more details.
Reflect and check

We could also have used a chart or an array to determine the possible outcomes.

1234
1(1,1)(1,2)(1,3)(1,4)
2(2,1)(2,2)(2,3)(2,4)
3(3,1)(3,2)(3,3)(3,4)
4(4,1)(4,2)(4,3)(4,4)

Regardless of which method we use, the sample space is the same.

A good check is that the size of the sample space is the number of possibilities in the first step times the number of possibilities in the second step.

Sample space size =4\cdot 4=16 which matches what we see in the tree diagram or array.

b

Are the events of selecting two cards independent or dependent?

Worked Solution
Create a strategy

We can use the fact that the cards are drawn with replacement to determine whether they are independent or dependent.

Apply the idea

From the tree diagram, we can see that the number of outcomes for both selections is the same. Because the first card is replaced, the outcome of the first draw will not affect the outcome of the second draw.

Therefore, the events are independent.

Idea summary

When selecting items with replacement, the first item selected is replaced before another item is selected. The first selection will have no effect on the second selection, so the events are independent.

When selecting items without replacement, the first item selected is not replaced before the second item is selected. The first selection will have an effect on the second selection, so the events are dependent.

Outcomes

8.PS.1

The student will use statistical investigation to determine the probability of independent and dependent events, including those in context.

8.PS.1a

Determine whether two events are independent or dependent and explain how replacement impacts the probability.

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