### Video Transcript

A bag contains an equal number of
blue, green, black, white, and red marbles. There are 30 marbles in the
bag. Four marbles are randomly taken out
of the bag. Part a) Work out the probability
of choosing four black marbles. The second part of the question
says some marbles are removed from the bag, such that there are still an equal
number of marbles of each colour and more than 15 marbles. Part b) Has the probability of
choosing four black marbles increased, decreased, or stayed the same? You must explain your answer.

As there are 30 marbles in total
and there are five different colours, we need to divide 30 by five. 30 divided by five is equal to
six. Therefore, there are six marbles of
each colour.

We are asked to calculate the
probability of four black marbles. The probability that the first
marble is black is six out of 30, as there are six black marbles and there are 30
marbles in total.

The probability can be calculated
by dividing the number of successful outcomes by the number of possible
outcomes. The probability that the second
marble is also black is five out of 29, as there are now five black marbles in the
bag and there are 29 in total. The probability that the third
marble is black is four out of 28, as we have now removed two black marbles. So there are four black ones left
and 28 left in total. And finally, the probability that
the fourth marble is also black is three out of 27. There are three black marbles left
in the bag, and there are 27 marbles altogether.

As we want all four of the marbles
to be black, we need to use the AND rule. This means that we will multiply
the fractions. We want the first marble to be
black and the second marble and the third marble and the fourth marble. Therefore, we multiply six out of
30 by five out of 29 by four out of 28 by three out of 27. Multiplying these four fractions
gives us an answer of one out of 1827. This means that there is a one in
1827 chance of selecting four black marbles.

The second part of our question
tells us that some marbles have been removed from the bag. But there are still an equal number
of each colour. And there are more than 15 marbles
altogether.

There are two possibilities
here. Either we could have 20 marbles
left in the bag, four of each colour, or alternatively we could have 25 marbles left
in the bag, five of each colour: five blue, five green, five black, five white, and
five red marbles.

We need to calculate the
probability of choosing four black marbles in both of these scenarios and comparing
our answers to our answer in part a. If there are four black marbles and
20 marbles altogether in the bag, the probability that the first marble is black is
four out of 20. The probability that the second
marble is black is three out of 19, the third marble being black would be two out of
18, and the fourth marble being black would be one out of 17.

Once again, we need to multiply all
four of these fractions. This gives us an answer of one out
of 4845. The probability of selecting four
black marbles is one out of 4845. We can use the same method to
calculate the probability of choosing four black marbles when there are five of each
colour and 25 marbles in total. We need to multiply five out of 25
by four out of 24 by three out of 23 by two out of 22. This gives us a probability of
selecting four black marbles of one out of 2530.

Both of these fractions are less
than one out of 1827. This means that the probability has
decreased. It is now less likely to select
four black marbles than it was in part a when there were 30 marbles in the bag.

It is important to note that if
there were 15 marbles in total in the bag, it would be impossible to select four
black marbles as there would only be three in the bag in total. Therefore, the probability of
choosing four black marbles would be zero.