Topic 1: The Fundamental Counting Principle

ProbabilityTopic 4A: Independent and Dependent Events Using the Product RuleI can compare, using examples, dependent and independent events.I can determine the probability of an event, given the occurrence of a previous event.I can determine the probability of two dependent or two independent events.I can solve a contextual problem that involves determining the probability of dependent or independent events.ExploreLeila and Caris each have 19 marbles: 11 red and 8 blue. Leila places 7 red marbles and 3 blue marbles in bag 1. She places the rest of her marbles in bag 2. Caris places all of her marbles in bag 3. Leila then draws one marble from bag 1 and one marble from bag 2. Caris draws 2 marbles from bag 3, without replacement.

Try the Explore on your own first. Then look at the solutions on the next slides.

Bag 1Bag 27 Red3 Blue4 Red5 Blue

Bag 311 Red8 BlueLeilas BagsCariss BagExplore1. Let A represent Leila drawing a red marble from bag 1.Let B represent Leila drawing a red marble from bag 2.Are A and B independent or dependent events? Explain.

2. Determine P(A).

3. Determine P(B).

Independent events the outcome of the 1st event does not affect the outcome of the second event.Dependent events the outcome of the 1st event does affect the outcome of the second event.

Bag 1Bag 27 Red3 Blue4 Red5 BlueLeilas BagsThese events are independent since the two draws are from different bags.Explore

The two individual probabilities can be multiplied. P(AB) = P(A) P(B)

Bag 311 Red8 BlueCariss BagThe events are dependent since the two draws are from the same bag without replacement.Explore6. Determine P(C) .

7. Determine the probability of event D, given that event C occurred. That is, what is the probability that Caris will draw a red marble from bag 3 on the second draw, given that the first marble she drew was red and was not replaced. This can be written as P(D|C).

Bag 311 Red8 BlueCariss BagExplore

The two individual probabilities can be multiplied. P(CD) = P(C) P(D|C)We can determine whether a situation is dependent or independent by the description. We must consider whether or not the draws are made from the same bag and whether or not the first draw is replaced before a second draw.InformationIndependent events are two events in which the outcome of the 1st event does not affect the outcome of the second event.Using the product rule for independent events, the probability that two independent independent events will both occur is the product of their individual probabilities:

InformationDependent events are two events in which the outcome of the 1st event does affect the outcome of the second event.Using the product rule for dependent events, the probability that two dependent events will both occur is the product of their individual probabilities:

P(B|A) is the probability that the second event, B, will occur, given that the first event, A, already occurred.

Example 1Classify the following events as either independent or dependent. Explain.a) rolling a 4 on a die and tossing heads on a coin

b) rolling a 2 on a die and rolling a 5 on a different die

c) rolling a 2 the first time on a die and rolling a 5 the second time on the same die

d) drawing a heart from a deck of cards, then drawing another heart from the same deck, without replacement

Classifying events as independent or dependentIndependent. These are 2 very separate events and do not affect one another.Independent. These are rolls on 2 different die, and they do not affect one another.Independent. Once die roll does not affect the next.Dependent. If you draw a heart from a deck and dont replace it, the second is affected by the absence of that card.Example 1e) drawing a black card from a deck of cards, then drawing a red card from the same deck, with replacement

f) picking a blue marble from one bag, then picking a purple marble from the same bag, with replacement

g) picking a blue marble from one bag, then picking a purple marble from a different bag, without replacement

Classifying events as independent or dependentIndependent. Since the first draw is returned to the deck before the second, the first draw does not affect the second.Independent. Since the 1st marble is replaced after being drawn, the 2nd draw is not affected by the 1st.Independent. Even though the 1st drawn marble was not replaced, it does not affect the 2nd draw. The 2nd draw is from a different bag.Example 2Mokhtar and Chantelle are playing a game that involves rolling a die and tossing a coin.a) Find the sample space for one die toss and one coin toss by drawing a tree diagram.

Determining probabilities of independent events1

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6HTHTHTHTHTHT1H1T2H2T3H3T4H4T5H5T6H6TExample 2Determining probabilities of independent events6 2 = 12Example 2Determining probabilities of independent events1

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6HTHTHTHTHTHT1H1T2H2T3H3T4H4T5H5T6H6TExample 3The probability that a student completes a math assignment is 0.8. The probability that the student completes an English assignment is 0.3. Assuming that these events are independent, find the following probabilities.a) the student completes both assignments

b) the student completes neither assignmentDetermining probabilities of independent eventsThe word neither means the complement of the 1st event and the complement of the 2nd event.Example 4Working backwards to determining probabilityThese events are independent!Example 5Two cards are drawn from a standard deck of cards. Compare the differences in how replacing the card after the first draw versus not replacing the card affects the probabilities of the events in the table. Calculate the probabilities using a formula.Comparing probabilities of independent and dependent eventsEventsWith ReplacementWithout Replacementboth cards are clubsa red jack first and then a black card secondExample 5Two cards are drawn from a standard deck of cards. Compare the differences in how replacing the card after the first draw versus not replacing the card affects the probabilities of the events in the table. Calculate the probabilities using a formula.Comparing probabilities of independent and dependent eventsEventsWith ReplacementWithout ReplacementA red jack and a black cardred jack firstblack card firstOrder is not specified here so we need to address each order separately.or0.0192 +0.0192 = 0.03850.0196 +0.0196 = 0.0392Example 7There is a bag of 20 marbles: 9 purple marbles, 3 blue marbles, 6 white marbles, and 2 green marbles. Two marbles are chosen, without replacing the first one.a) Are these events independent or dependent? Does it matter if the first marble is replaced or not? Explain.

Determining the probability There events are dependent, since the marbles are drawn from the same bag with no replacement.Example 7b) Find the following probabilities, if possible:i) drawing 2 purple marbles ii) drawing no purple marbles)

Determining the probability Example 7b) Find the following probabilities, if possible:iii) drawing a purple marble and then a white marble

iv) drawing a purple marble and a white marble

Determining the probability or

Need to KnowIndependent events are two events in which the outcome of the 1st event does not affect the outcome of the second event.Using the product rule for independent events, the probability that two independent events will both occur is the product of their individual probabilities:

Need to KnowDependent events are two events in which the outcome of the 1st event does affect the outcome of the second event.Using the product rule for dependent events, the probability that two dependent events will both occur is the product of their individual probabilities:

Need to KnowA tree diagram is often useful for modeling problems that involve independent or dependent events.The word neither means the complement of the 1st event and the complement of the 2nd event.Sometimes replacing or not replacing an item before choosing a second item can affect whether the two events are independent or dependent.

Youre ready! Try the homework from this section.Replace 1st itemDo NOT Replace 1st itemItems come from the same containerIndependentDependentItems do NOT come from the same containerIndependentIndependent