Independent and Dependent Events

Learning Targets Determine when events are dependent or independent. Learn to use the multiplication rule of probability correctly.

Question A dresser drawer contains one pair of socks with each of the following colors: blue, brown, red, white and black. Each pair is folded together in a matching set. You reach into the sock drawer and choose a pair of socks without looking. You replace this pair and then choose another pair of socks. What is the probability that you will choose the red pair of socks both times?

Look Familiar? Where have we seen similar problems like this “sock drawer” problem? Spinning a Spinner Multiple Times Rolling Dice Flipping Coins These are all examples of compound events where we are looking for the probability of specific multiple events occurring.

Compound Events We can determine the probability of compound events happening by applying the multiplication rule. However, in order to apply this rule properly it must be determined whether the compound events are independent or dependent.

Definitions Independent Events: Two events, A and B, are independent if the fact that A occurs does not affect the probability of B occurring. Dependent Events: Two events are dependent if the outcome or occurrence of the first affects the outcome or occurrence of the second so that the probability is changed.

Back to the Question… A dresser drawer contains one pair of socks with each of the following colors: blue, brown, red, white and black. Each pair is folded together in a matching set. You reach into the sock drawer and choose a pair of socks without looking. You replace this pair and then choose another pair of socks. What is the probability that you will choose the red pair of socks both times? Is this a dependent or independent event?

Why does it matter… Independent and dependent events will impact how probabilities of compound events are calculated. Ex: Look at the following two scenarios Drawing a king, replacing it, and then drawing another king. Drawing a king and then drawing another king.

Slight Differences These two events seem like they are asking the same question. However, they have different odds of occurring. The first example allows us to replace the card so the odds in both pulls are the same, the deck never changes. In the second example we no longer replaced the first king drawn. This leaves us with only 3 kings remaining out of 51 cards.

Independent vs. Dependent Events These slight differences can start to add up and really effect the probabilities of outcomes. It is imperative to determine whether events are independent or dependent before calculating the probability of a compound event.

Independent Events With independent events our probabilities of compound events do not effect one another. This allows us to use the multiplication rule for independent events:

Dependent Events With dependent events our probabilities of compound events do effect one another. This means we must use the multiplication rule for dependent events: Probability of event B given A

Practice: Example 1: Ms. Winner needs two students to help her with a science demonstration for her class of 18 girls and 12 boys. She randomly chooses one student who comes to the front of the room. She then chooses a second student from those still seated. What is the probability that both students chosen are girls?

Practice: Example 2: A coin is tossed and a single 6-sided die is rolled. Find the probability of landing on the head side of the coin and rolling a 3 on the die.

Practice: Example 3: A school survey found that 9 out of 10 students like pizza. If three students are chosen at random with replacement, what is the probability that all three students like pizza?

Practice: Example 4: In a shipment of 20 computers, 3 are defective. Three computers are randomly selected and tested. What is the probability that all three are defective if the first and second ones are not replaced after being tested?

Practice: Example 5: In a shipment of 20 computers, 3 are defective. Three computers are randomly selected and tested. What is the probability that all three are non-defective if the first and second ones are not replaced after being tested?

Large Sample Spaces All of our examples have had smaller sample spaces where changes in the odds have large impacts. For large scale sample spaces we can treat the probability of compound events occurring as if they are all independent. Example A nationwide survey found that 72% of people in the United States like pizza. If 3 people are selected at random, what is the probability that all three like pizza?