5.01 Introduction to independent and dependent events
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.
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
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
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 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.
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.
Example 2
This spinner is spun and a six-sided die is rolled.
Create an array that shows the sample space for the experiment.
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?
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 red marble is drawn from the jar, then placed back into the jar. How many marbles are in the jar?
A green marble is removed from the jar. How many marbles are left in the jar?
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.
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.
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.
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.
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.
Are the events of selecting a captain and a co-captain independent or dependent?
Example 4
Four cards numbered 1 to 4 are placed face down on a table. Two cards are drawn with replacement.
Construct a tree diagram of this situation.
Are the events of selecting two cards independent or dependent?
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.