1. Topic 4A: Independent and Dependent Events Using the Product Rule
Probability

2. I 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.

3. Explore…
Try the Explore on your own first. Then look at the solutions on the next slides.
Leila 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.
Leila’s Bags
Caris’s Bag
7 Red
3 Blue
4 Red
5 Blue
11 Red
8 Blue
Bag 1
Bag 2
Bag 3

4. 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.
Explore…
1. 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).
These events are independent since the two draws are from different bags.
P(A) = 7 10
Leila’s Bags
P(B) = 4 9
7 Red
3 Blue
4 Red
5 Blue
Bag 1
Bag 2

5. Explore…
4. The probability that both events A and B occur is This can be written as P(A and B) or P(A∩B) . How can you use the probability of A and the probability of B to get the probability of A and B? 5. Let C represent Caris drawing a red marble from bag 3 on her first draw and keeping this marble. Let D represent Caris drawing a red marble from bag 3 on her second draw. Are C and D independent or dependent events? Explain.
The two individual probabilities can be multiplied. P(A∩B) = P(A) × P(B)
Caris’s Bag
11 Red
8 Blue
The events are dependent since the two draws are from the same bag without replacement.
Bag 3

6. Caris’s Bag
Explore…
11 Red
8 Blue
Bag 3
6. 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).
P(C) =
P(D|C) =

7. Explore…
8. The probability that both events C and D occur is This can be written as P(C and D) or P(C∩D). How can you use P(C) and P(D|C) to get the probability of C and D? 9. Explain how you can tell if two events are independent or dependent.
The two individual probabilities can be multiplied. P(C∩D) = 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.

8. Information
Independent 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:

9. Information
Dependent 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.

10. Example 1
Classifying events as independent or dependent
Classify 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
Independent. 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 don’t replace it, the second is affected by the absence of that card.

11. Example 1
Classifying events as independent or dependent
e) 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
Independent. 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.

12. Example 2
Determining probabilities of independent events
Mokhtar 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. H T 1H 1T 2H 2T 3H 3T 4H 4T 5H 5T 6H 6T 1 2 3 4 5 6

13. Example 2
Determining probabilities of independent events
b) Find the number of outcomes in the sample space. c) Use the sample space from the tree diagram to find P(6∩H).
6 × 2 = 12
= 1 12

14. Example 2
Determining probabilities of independent events
1 2 H T 1H 1T 2H 2T 3H 3T 4H 4T 5H 5T 6H 6T 1 2 3 4 5 6
d) Label each branch in the tree diagram with its probability. Use this new information (and the product rule) to find P(6∩H). Do not use the sample space.
1 6
= × = 1 12

15. The word “neither” means the complement of the 1st event and the complement of the 2nd event.
Example 3
Determining probabilities of independent events
The 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 assignment
P(M ∩ E) = 0.8 × 0.3
P(M ∩ E) = 0.24
P(M’ ∩ E’) = 0.2 × 0.7
P(M’ ∩ E’) = 0.14

16. These events are independent!
Example 4
Working backwards to determining probability
The probability that the Oilers will win a championship is The probability that the Oilers and the Eskimos will both win a championship is What is the probability that the Eskimos will win a championship?
P(Oilers win ∩ Eskimos win) = P(Oilers win) × P(Eskimos win)
8 63 = × P(Eskimos win)
P(Eskimos win) =
8 63 ÷ = P(Eskimos win)
P(Eskimos win) = 4 7
8 63 × = P(Eskimos win)

17. Example 5
Comparing probabilities of independent and dependent events
Two 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.
Events
With Replacement
Without Replacement
both cards are clubs
a red jack first and then a black card second
× = =
× = =
2 52 × = =
2 52 × = =

18. Example 5 Comparing probabilities of independent and dependent events
Two 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.
Events
With Replacement
Without Replacement
A red jack and a black card
Order is not specified here so we need to address each order separately.
2 52 × = =
2 52 × = =
red jack first or
× = =
× = =
black card first
0.0192… … =
0.0196… … =

19. Example 7
Determining the probability
There 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.
There events are dependent, since the marbles are drawn from the same bag with no replacement.

20. Example 7 b) Find the following probabilities, if possible:
Determining the probability
b) Find the following probabilities, if possible:
 i) drawing 2 purple marbles ii) drawing no purple marbles)
P(P ∩ P) = P(P) × P(P|P)
P(P ∩ P) = × 8 19
P(P ∩ P) =
P(P’ ∩ P’) = P(P’) × P(P’|P’)
P(P ∩ P) = ×
P(P ∩ P) =

21. Example 7 b) Find the following probabilities, if possible:
Determining the probability
b) 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
P(P ∩ W) = P(P) × P(W|P)
= × 6 19 =
P(P ∩ W) = P(P) × P(W|P)
= × 6 19 =
P(W ∩ P) = P(W) × P(P|W)
= × 9 19 = or
= 𝑜𝑟

22. Need to Know
Independent 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:

23. Need to Know
Dependent 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:

24. Need to Know
You’re ready! Try the homework from this section.
A 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.
Replace 1st item
Do NOT Replace 1st item
Items come from the same container
Independent
Dependent
Items do NOT come from the same container