Gr 11 Probability: Tree diagram
Summary
TLDRThis 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
🎨 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.
🔢 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.
Mindmap
Keywords
💡Tree Diagram
💡Probability
💡Outcomes
💡Blue Marble
💡Orange Marble
💡Branches
💡Multiplication Rule
💡Addition Rule
💡Complementary Events
💡Independent Events
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
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