Contributor: Mason Smith. Lesson ID: 11310

How well do you understand probability using multiple events? It all DEPENDS on how you complete this lesson! Learn about dependent and independent events with online practice and some observations!


Middle School

learning style
personality style
Grade Level
Middle School (6-8), High School (9-12)
Lesson Type
Quick Query

Lesson Plan - Get It!


Q: What do you call two unrelated activities scheduled on July 4th?

A: Independent events.

hus far in this Probability: An Overview series, found under Related Lessons in the right-hand sidebar, we have covered experimental and theoretical probability.

However, the probability we have covered so far has only been for one single event. What if we wanted to calculate the probability of multiple events? This is where we have to take into account the idea of dependency, or how each event influences the other.

For example, if we roll a number cube and pull an ace from a standard card deck, do these two experiments have any effect on one another? Of course not! It doesn't matter what you roll, you will still have the same probability of pulling an ace from a deck. This is the idea of independent events, when the first event does not have an effect on the probability of the other event.

The other form of dependency is a dependent event, when the first event affects the probability of the second event. If we have an experiment of pulling one card from a deck and then another card from the deck, what is the probability that the second card will be an ace? It is no longer 4/52, since there are not 52 cards in the deck (we already took one out), so the probability of the second card being an ace is 451.

Classify the following two events as either independent or dependent. Discuss your answers with a teacher or parent:

  • You flip a penny, then flip a nickel to get heads.
  • Answer: Independent, because no matter what the penny landed on, the nickel still has ½ probability of landing on heads.
  • Your sister selects a Monopoly piece from the 8 available, then you pick one from the remaining.
  • Answer: Dependent, because once your sister selects a piece, there are fewer remaining pieces for you to choose from.

When we are working with dependent events, we can construct a tree diagram to keep track of the possibilities. Here is an example of a tree diagram for flipping a coin twice. When we calculate this, we want to work from left to right. So, if we are trying to find the probability of getting {HT}, then we multiply ½ (the probability for heads the first time which is on the "branch") times ½ (probability of tails the second time) to get the probability of {HH}, which is ¼.

This leads to an important formula for independent events:

If A and B are independent events, then P(A and B)= P(A) * P(B).

So, the probability of both A and B happening is the product of each event.

Let's try this out with an example:

  • An experiment consists of randomly selecting a marble from a bag, replacing it, and then selecting another marble. The bag has 7 blue marbles and 3 yellow marbles. Since the marbles are immediately replaced, the probability does not change and the events are independent. What is the probability of pulling out a blue, then a yellow marble? Discuss your answers with a teacher or parent.
P(blue, yellow) =P(blue) * P(yellow) =  7  +  3  =  21 
10 10 100


  • What is the probability of spinning two odd numbers? Discuss your answers with a teacher or parent:

There are 8 results, with 4 odd results, so the probability of spinning an odd once is ½. It is independent because the first spin has no effect on the second.

P(2 odd numbers) =  1  *  1  =  1 
2 2 4


Now that we know how to calculate independent events, let's try calculating dependent probability. Remember, with independent events, the probability does not change, but for dependent events, the probability will change from the first event to the second. The formula for dependent events looks like this:

P(A and B) = P(A) * P(B after A)

However, that doesn't make much sense just looking at it, so let's try it with an example:

  • An experiment consists of randomly selecting a marble from a bag, NOT replacing it, then selecting another marble. The bag has 4 blue marbles and 6 yellow marbles. What is the probability of selecting a blue then a yellow marble? Discuss your answers with a teacher or parent.

Well, since we do not replace the marble, these two events are dependent events. The probability of selecting a blue is pretty easy, so we can set up our formula like this:

P(Blue, Yellow) = P(Blue) * P(Y after B)

But, what is P(Y after B)? That is the probability of pulling a yellow, after we have already removed a blue, which means our probability is:

P(Y after B) =                    # of yellow                          =  6 
# of total results after removing one 9


Now, we can solve for the probability of pulling a blue then a yellow rather easily. Give it a try, then look at the correct result.

P(Blue, Yellow) =  4  *  6  =  24  =  12 
10 9 90 45


Now that you have learned about the idea of dependency, independent, and dependent events, it is time to practice your new skills.

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