SOL Objective(s):
8.12 The student will determine the probability of independent and dependent events with and without replacement.
The Basics and Background Information
• Two events are either dependent or independent.
• If the outcome of one event does not influence the occurrence of the other event, they are called independent. If events are independent, then the second event occurs regardless of whether or not the first occurs. For example, the first roll of a number cube does not influence the second roll of the number cube. Other examples of independent events are, but not limited to: flipping two coins; spinning a spinner and rolling a number cube; flipping a coin and selecting a card; and choosing a card from a deck, replacing the card and selecting again.
• The probability of three independent events is found by using the following formula: P(Aand Band C) = P(A) x P(B) x P(C)
Ex: When rolling three number cubes simultaneously, what is the probability of rolling a 3 on one cube, a 4 on one cube, and a 5 on the third?
P(3 and 4 and 5) = P(3) x P(4) x P(5) = 1/6 x 1/6 x 1/6 = 1/216
• If the outcome of one event has an impact on the outcome of the other event, the events are called dependent. If events are dependent then the second event is considered only if the first event has already occurred. For example, if you are dealt a King from a deck of cards and you do not place the King back into the deck before selecting a second card, the chance of selecting a King the second time is diminished because there are now only three Kings remaining in the deck. Other examples of dependent events are, but not limited to: choosing two marbles from a bag but not replacing the first after selecting it; and picking a sock out of a drawer and then picking a second sock without replacing the first.
• The probability of two dependent events is found by using the following formula: P(Aand B) = P(A) x P(Bafter A)
Ex: You have a bag holding a blue ball, a red ball, and a yellow ball. What is the probability of picking a blue ball out of the bag on the first pick then without
replacing the blue ball in the bag, picking a red ball on the second pick? P(blue and red) = P(blue) x P(red after blue) = 1/3 x 1/2 = 1/6
• If the outcome of one event does not influence the occurrence of the other event, they are called independent. If events are independent, then the second event occurs regardless of whether or not the first occurs. For example, the first roll of a number cube does not influence the second roll of the number cube. Other examples of independent events are, but not limited to: flipping two coins; spinning a spinner and rolling a number cube; flipping a coin and selecting a card; and choosing a card from a deck, replacing the card and selecting again.
• The probability of three independent events is found by using the following formula: P(Aand Band C) = P(A) x P(B) x P(C)
Ex: When rolling three number cubes simultaneously, what is the probability of rolling a 3 on one cube, a 4 on one cube, and a 5 on the third?
P(3 and 4 and 5) = P(3) x P(4) x P(5) = 1/6 x 1/6 x 1/6 = 1/216
• If the outcome of one event has an impact on the outcome of the other event, the events are called dependent. If events are dependent then the second event is considered only if the first event has already occurred. For example, if you are dealt a King from a deck of cards and you do not place the King back into the deck before selecting a second card, the chance of selecting a King the second time is diminished because there are now only three Kings remaining in the deck. Other examples of dependent events are, but not limited to: choosing two marbles from a bag but not replacing the first after selecting it; and picking a sock out of a drawer and then picking a second sock without replacing the first.
• The probability of two dependent events is found by using the following formula: P(Aand B) = P(A) x P(Bafter A)
Ex: You have a bag holding a blue ball, a red ball, and a yellow ball. What is the probability of picking a blue ball out of the bag on the first pick then without
replacing the blue ball in the bag, picking a red ball on the second pick? P(blue and red) = P(blue) x P(red after blue) = 1/3 x 1/2 = 1/6
Essential Knowledge and Skills
The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to...
• Determine the probability of no more than three independent events.
• Determine the probability of no more than two dependent events without replacement.
• Compare the outcomes of events with and without replacement.
• Determine the probability of no more than three independent events.
• Determine the probability of no more than two dependent events without replacement.
• Compare the outcomes of events with and without replacement.
Vocabulary and Things to Know:list of info

Be Able To:list of info

The topic: What is it and what will we learn?
Independent Events are when nothing is removed before second or another event.
Dependent Events are when something IS removed and not replaced before another event.
Dependent Events are when something IS removed and not replaced before another event.
Independent AND Dependent EventsThe same objects are used to demonstrate independent events, as well as dependent events.


Probability of Dependent Events 
