Gr 11 Probability: Tree diagram

Summary

This educational script explains the concept of tree diagrams in probability through a scenario where Kate draws marbles from a bag containing 3 orange and 7 blue marbles. It illustrates how to construct a tree diagram, calculate probabilities for different outcomes like drawing two oranges or two blues, and determine the chance of drawing at least one blue or orange marble. The script simplifies complex probability calculations by using the tree diagram method, making it accessible and engaging for learners.

Takeaways

• 🎲 Kate draws marbles from a bag containing 10 marbles, with 3 being orange and 7 being blue.
• 🌳 A tree diagram is used to visualize the probability of drawing different colored marbles, illustrating the process of drawing twice.
• 🔵 The probability of drawing a blue marble on the first draw is 7 out of 10, and it remains the same for the second draw due to replacement.
• 🟠 The probability of drawing an orange marble on the first draw is 3 out of 10, and it also remains the same for the second draw.
• 🍊 The probability of drawing two orange marbles is calculated by multiplying the probabilities of each individual draw (3/10 * 3/10 = 9/100).
• 🔵 The probability of drawing two blue marbles is found by multiplying the probabilities of drawing blue on both draws (7/10 * 7/10 = 49/100).
• 🤔 The probability of drawing at least one blue marble is calculated by adding the probabilities of all branches that include a blue marble (49/100 + 21/100 + 21/100 = 91/100).
• 🧩 A faster method to find the probability of drawing at least one blue is to subtract the probability of drawing no blue marbles from 1 (1 - 9/100 = 91/100).
• 🍊 The probability of drawing at least one orange marble is the sum of the probabilities of the branches that include at least one orange marble (21/100 + 9/100 = 51/100).
• 📊 Understanding the concept of 'and' and 'or' in probability is crucial; 'and' requires multiplication of probabilities, while 'or' requires addition.

Q & A

What is the total number of marbles in the bag according to the transcript?

There are ten marbles in the bag, with three being orange and the rest being blue.

How does the process of drawing marbles with replacement work as described in the transcript?

Kate draws one marble, records its color, puts it back into the bag, and then draws a second marble. This is called drawing with replacement because the first marble is returned to the bag before the second draw.

What is the probability of drawing a blue marble on the first draw?

The probability of drawing a blue marble on the first draw is 7 out of 10, since there are seven blue marbles out of a total of ten.

What is the probability of drawing an orange marble on the second draw if the first marble drawn was blue?

If the first marble drawn was blue and put back, the probability of drawing an orange marble on the second draw remains the same at 3 out of 10.

How is the tree diagram used to illustrate the probabilities of drawing marbles?

The tree diagram is used to visualize the different outcomes and their probabilities for each draw. It branches out to show the possible combinations of drawing blue or orange marbles for each draw.

What is the probability of Kate drawing two orange marbles in a row?

The probability of drawing two orange marbles in a row is calculated by multiplying the probability of drawing an orange marble on the first draw (3 out of 10) by the probability of drawing an orange marble on the second draw (also 3 out of 10), which equals 9 out of 100 or 0.09.

How is the probability of drawing two blue marbles calculated?

The probability of drawing two blue marbles is calculated by multiplying the probability of drawing a blue marble on the first draw (7 out of 10) by the probability of drawing a blue marble on the second draw (7 out of 10), resulting in 49 out of 100 or 49%.

What is the significance of the term 'and' in calculating probabilities in the context of the transcript?

In the context of the transcript, the term 'and' indicates that the probabilities of independent events should be multiplied together to find the probability of both events occurring.

How does the transcript suggest a faster way to calculate the probability of drawing at least one blue marble?

The transcript suggests a faster way by calculating the probability of the complementary event (drawing two orange marbles) and then subtracting it from 1 to find the probability of drawing at least one blue marble.

What is the probability of Kate drawing at least one orange marble, and how is it calculated?

The probability of drawing at least one orange marble is calculated by adding the probabilities of the branches that include at least one orange marble (blue-orange, orange-blue, and orange-orange), which equals 51 out of 100. Alternatively, it can be calculated by subtracting the probability of drawing two blue marbles (9 out of 100) from 1.

Outlines

00:00

🎨 Understanding Tree Diagrams for Probability

This paragraph introduces the concept of tree diagrams as a tool for visualizing probabilities in a scenario where Kate draws marbles from a bag containing a mix of blue and orange marbles. The paragraph explains the process of creating a tree diagram by considering the probabilities of drawing a blue or orange marble on two separate occasions. It emphasizes that tree diagrams are a straightforward method for understanding probability, despite initial apprehensions. The paragraph also discusses how to calculate the probability of specific outcomes, such as drawing two orange marbles, by multiplying the probabilities of each draw, given that the marbles are replaced after each draw.

05:00

🔢 Calculating Probabilities of Marble Draws

The second paragraph delves into calculating the probability of drawing two blue marbles, which is higher due to the greater number of blue marbles in the bag. It explains the process of determining the probability of drawing at least one blue marble by adding the probabilities of the different branches that result in a blue marble being drawn. The paragraph introduces a shortcut for calculating probabilities by using the total probability of all outcomes (which should sum to 1 or 100%) and subtracting the probability of the undesired outcome to find the probability of the desired outcome. This method is illustrated through the calculation of the probability of drawing at least one orange marble, which is done by subtracting the probability of drawing two orange marbles from the total probability.

Keywords

💡Tree Diagram

A tree diagram is a graphical representation used to illustrate the possible outcomes of a sequence of events. In the context of the video, it is used to visually represent the probability of drawing marbles from a bag. The script describes how to create a tree diagram for Kate's two draws from a bag containing marbles, showing the branches for each possible outcome (drawing a blue or orange marble) and how these outcomes can be combined to determine the probability of various scenarios.

💡Probability

Probability is the measure of the likelihood that a particular event will occur. The video script discusses how to calculate probabilities using a tree diagram, particularly in the context of drawing marbles from a bag. It explains how the probability of drawing a blue or orange marble is represented and how these probabilities are used to determine the likelihood of different outcomes, such as drawing two orange marbles or at least one blue marble.

💡Outcomes

Outcomes refer to the possible results of an event. In the video, the script describes the two possible outcomes when drawing a marble from the bag: drawing a blue marble or an orange marble. These outcomes are the basis for constructing the tree diagram and calculating probabilities.

💡Blue Marble

A blue marble is one of the two types of marbles in the bag, as described in the script. There are seven blue marbles, making them the more probable outcome when drawing a single marble. The script uses the blue marble as an example to explain how to calculate the probability of drawing two blue marbles or at least one blue marble in two draws.

💡Orange Marble

An orange marble is the less common type of marble in the bag, with only three present. The script uses the orange marble to illustrate the calculation of the probability of drawing two orange marbles, which is a less likely event compared to drawing blue marbles due to the lower number of orange marbles.

💡Branches

In the context of a tree diagram, branches represent the different paths or sequences of outcomes that can occur. The script explains how each draw of a marble creates a branch, and how these branches are used to calculate the probability of various combinations of outcomes, such as 'blue-blue', 'blue-orange', etc.

💡Multiplication Rule

The multiplication rule in probability states that if two events are independent, the probability of both events occurring is the product of their individual probabilities. The script uses the multiplication rule to calculate the probability of drawing two orange marbles, where the probability of drawing an orange marble on the first draw is multiplied by the probability of drawing an orange marble on the second draw.

💡Addition Rule

The addition rule in probability is used to combine the probabilities of two or more mutually exclusive events. The script mentions that to find the probability of drawing at least one blue marble, one would add the probabilities of the different branches that result in drawing a blue marble, such as 'blue-blue', 'blue-orange', and 'orange-blue'.

💡Complementary Events

Complementary events are two events where one event occurring means the other cannot. In the script, the complementary event to drawing two orange marbles is drawing at least one blue marble. The video explains how to find the probability of complementary events by subtracting the probability of the less likely event (drawing two oranges) from 1, which represents the total probability of all possible outcomes.

💡Independent Events

Independent events are events where the outcome of one does not affect the outcome of the other. The script implies that each draw of a marble is an independent event because replacing the marble after the first draw ensures the probabilities remain the same for the second draw.

Highlights

Introduction to creating a tree diagram for probability

Explanation of the marble drawing scenario with replacement

Simplification of drawing probabilities using ratios instead of multiple branches

Calculation of the probability for drawing two orange marbles

Understanding the 'and' condition in probability for multiplying probabilities

Determining the probability of drawing two blue marbles

Explanation of the 'or' condition in probability for adding probabilities

Calculating the probability of drawing at least one blue marble

Using subtraction to find the probability of drawing at least one blue marble

Highlighting the importance of total probabilities summing to 100%

Determining the probability of drawing at least one orange marble

Using the complement rule to simplify probability calculations

Visual representation of the tree diagram with different branches

Explanation of the different outcomes for each branch of the tree diagram

Calculating the probability for each branch involving blue and orange marbles

Summing up probabilities for branches to find the total chance of drawing a blue marble

Providing a faster method for calculating probabilities by using the complement

Transcripts

so in this question we have a bag that

contains

ten marbles of which three are orange

and the rest are blue we are told that

kate

will draw one marble so she'll take one

marble out of the bag

and then she'll put it back and then

she'll take out

a second marble the first question says

draw a tree diagram to illustrate the

above information

so a lot of people panic with tree

diagrams but i promise you they're

actually probably

they're probably one of the easiest

parts of probability

so what you've got to do is imagine

yourself as kate okay so we're going to

put a little dot

over here which will mark the start of

our

tree diagram now if you were kate

and you walk up to that bag for the very

first time

you have okay now let's say you're not

you're not looking in the bag

so you stick your hand into the bag and

you choose something what are the two

possible outcomes you could either pull

out

a blue marble or you could pull out

an orange now of course you could go

draw

seven blue arrows and you could go draw

three orange ones so you'd have a total

of ten arrows but that's complicated

what we rather do is the following we

will say that the probability of a blue

well out of a total of ten seven of them

were blue and out of a total of ten our

chances of getting an orange

were three now you have to imagine

which one so let's imagine that kate

chose

blue okay so she's gone this way

now you completely forget about

this part we'll get back to that later

so now kate

is about okay so she's taken that marble

out

she's put it back and now she's about to

do her second attempt

so when she puts her hand into the bag

now what are her possibilities

well once again she could draw a blue

or of course she could draw orange and

her probability well for the blue

well there's still 10 marbles left in

the bag because she put the other one

back and so they're still going to be

seven blue ones

and the probability for orange would

still be three out of ten

if however and we'll look at this in a

future question

kate did not put the other marble back

then there would only be nine marbles

left and we would have to change things

up a bit

now let's imagine instead that kate went

down this path

the very first time well then when she

does her next

draw these are her two possibilities

and once again the possibilities or the

probabilities would be

7 out of 10 and 3 out of

10. now that everything's complete

we need to look at the different

combinations so if kate

did this that would be a blue marble

and another blue marble so i'm going to

call this the blue blue branch

likewise this will be the blue orange

blonde the branch not blanche

this would be the orange blue and this

would be the

orange orange so now the next question

says determine the probability that kate

draws two orange marbles okay so her

first one

is three out of ten and the next one is

three out of ten

so what would go in between that would

you say or

or would it be and does she draw an

orange or

another orange or does she draw an

orange and another orange

well she draws an orange and another

orange and so we're gonna multiply over

here

if it was or then we would use the or

formula remember we looked at this in

one of our previous videos that if it

says and

you have to multiply if it's or you

you plus so to memorize this whenever

you work

on a tree diagram question to work out

the probability of a specific branch

you multiply and so if we had to

multiply these now we would get a total

of 9 over 100 number three determine the

probability that kate

draws two blue marbles so that's the bb

branch

so that's seven over a hundred

multiplied

not a hundred seven over ten sorry so

it's that one

multiplied by this one which is seven

over ten

and that's going to give us 49 out of a

hundred

so her chances of getting two blues is

49 percent

and her chances of getting two

oranges was 0.09 or so it was 9 over 100

which is about 9

so her chances of drawing two blues are

much higher and that makes sense because

there's more blues

in the bag than orange number three

determine the probability that kate

draws a blue marble well that's

quite a lot to ask a blue marble could

be

any one of the following branches this

one because that's

bb so that's blue blue she could do this

one which is the blue orange

or she could do that one that's the

orange blue so we would have to go

add all of those together okay so let's

do that and then i'll show you a faster

way

so we already know the bb or the blue

blue branch that's 49 over 100 let's

quickly work out the blue

orange branch which is this one well

that's going to be 7

over 10 times by 3 over 10 which is 21

over 100

next we could look at the orange blue

where that will be 21

over 100 and then for the last one which

was the orange orange we already said

that that's 9

over 100 oh no but we don't want that

one anyway so we only want those first

three

now what's very important is that when

you take these three values now

now we can add you don't have to you

mustn't multiply these

these ones you are actually going to add

together

and that gives us a 91 out of 100

chance and that's 91 percent now that

makes sense

imagine you kate and you busy drawing

marbles out of a bag and you do this

twice

the chances that you get a blue are

going to be pretty high i mean

there's 77 marbles of blue and only

three orange

so you have a 91 chance that you would

get at least one blue

so kevin you mentioned that there was a

faster way to do this well

yes we know that if you had well the way

it works in

probability is that if you have to add

each of these different branches

together

which is all the total outcomes you

should get 100 over 100 which is

one or a hundred percent okay because

you yeah there's a if you if you

complete everything

it always equals a hundred percent for

example if we have venn diagrams

and let's say those two circles

completely give us

65 then on the outside we would have 35

percent

because we always have to end up with a

hundred percent if we're busy with

percentages

or one if we're busy with probability

fractions

so instead of adding this whole branch

plus this whole branch plus this whole

branch

why don't we just do this why don't we

just calculate

this branch which is nine over a hundred

and then just say or that equals 0.09

now we know that all of the probability

should add up to 1. so we could say

1 minus that branch which is 0.09

and that gives us 0.91 which is the same

as 91 over 100 which is what we got over

here

so instead of calculating three

different branches we only calculated

one

and then just subtract so let me explain

that once more

all four of those branches should add up

to

one which is a hundred over a hundred

the bottom branch is nine out of a

hundred so if that's nine out of a

hundred then it means that these three

branches over here

should give us ninety one out of

a next question says determine the

probability that kate

draws at least one orange

so at least one orange will that means

one orange or more so that would be this

branch

and this one and this one because

remember

the word at least means that

or more so you can have more so once

again we could go add each of these

together

which would give us 51 out of 100

or because we know that these four

branches

should equal 100 over 100 then we could

just say 100 over 100

minus this one over here and that also

gives us

51 over 100 so whichever way is best for

you