There are 13 marbles in a bag. Three of the marbles are red and
the rest are black. Sasha randomly takes out two
marbles from the bag without replacement. Work out the probability that the
two marbles are the same colour.
The word without is written in bold
type, which means it must be pretty important. If Sasha takes the marbles from the
bag without replacement, it means she doesn’t put the first marble back in the bag
before she selects the second. This means that there are only 12
marbles available for her second choice. That’s 13 minus one because she’s
already taken one marble out of the bag and not put it back. It’s really important that we
remember this as we go through the question.
We can use a tree diagram to help
us answer this question. We’ll start off with Sasha’s first
marble, which can be either red or black. Then, we have the second marble,
which can also be either red or black. Now let’s think about the
probabilities for each of these branches on the tree diagram.
To start off with, there are 13
marbles in the bag and three of them are red. Sasha chooses her marbles at
random, which means the probability of her selecting each marble is the same. So the probability that she chooses
a red marble is three out of 13. We’re told that the rest of the
marbles are black. So this means that there are 10
black marbles in the bag at the start. That’s 13 minus three. So the probability that the first
marble Sasha chooses is black is 10 out of 13.
What about the branches on the
second stage of the tree diagram? Remember Sasha doesn’t put the
first marble back in the bag, which means that there are now only 12 marbles
available to be chosen. So the denominators for the
fractions on the second set of branches are all going to be 12. If the first marble that Sasha took
was red, then there’re only two red marbles left in the bag. So the probability that the second
marble will be red if the first was red is two twelfths, two out of 12. But we haven’t taken any black
marbles out of the bag in this case. So there were still the 10 black
marbles that we started off with, which means the probability that Sasha’s second
marble is black if the first marble was red is 10 out of 12.
Finally, let’s consider what
happens if the first marble that Sasha took was black. There are still the same number of
red marbles in the bag as there were to begin with. So the probability that Sasha’s
second marble is red if the first marble was black is three out of 12. But this time, there are only nine
black marbles left because Sasha took one of them out of the bag. So the probability that her second
marble is black if the first marble was black is nine out of 12. Notice that the probabilities on
every set of branches on this tree diagram sum to one.
Now the question asked us to work
out the probability that the two marbles Sasha chooses are the same colour. There are two ways that Sasha could
do this. She could either get two red
marbles or she could get two black marbles. To find the probabilities at the
end of the branches on a tree diagram, we multiply together the probabilities along
the route that we need to take to get there.
So to find the probability that the
first marble is red and the second marble is red, we multiply three over 13 — that’s
the probability the first marble is red — by two over 12 — that’s the probability
the second marble is red if the first marble was red. Now let’s simplify this calculation
slightly. We can cross cancel a factor of
three from the three in the numerator of the first fraction and the 12 in the
denominator of the second, giving one and four. We can also cancel within the
second fraction. Two and four can both be divided by
two, giving one in the numerator and two in the denominator.
To multiply the two fractions
together, we multiply the numerators and then multiply the denominators. One multiplied by one is one and 13
multiplied by two is 26. So the probability that both
marbles that Sasha chooses are red is one over 26.
To find the probability that both
marbles are black, we multiply the probabilities along these branches of the tree
diagram. That’s 10 over 13 multiplied by
nine over 12. We can simplify the second fraction
by dividing both the numerator and denominator by three, giving three over four. And we can cross cancel a factor of
two from the 10 in the numerator of the first fraction and the four in the
denominator of the second, leaving five over 13 multiplied by three over two. Five multiplied by three is 15 and
13 multiplied by two is 26. So the probability that the two
marbles Sasha chooses are both black is 15 over 26.
Finally, we need to combine these
two probabilities together. And to do so, we use the OR rule of
probability, which tells us that if two events 𝐴 and 𝐵 are mutually exclusive,
meaning they can’t happen at the same time, then to find the probability of one
event or the other happening, we add their individual probabilities together. So to find the probability that
Sasha gets two red marbles or two black marbles, we add one over 26 and 15 over
To add fractions, we need a common
denominator which we already have. So then, we just add the two
numerators together, giving 16 over 26. However, we can simplify this
fraction by dividing both the numerator and denominator by two, giving eight over
13. This fraction can’t be simplified
any further. And there’s no need to try and
convert it to a decimal.
So we found that the probability
that the two marbles are the same colour is eight over 13.