# Video: Pack 1 • Paper 2 • Question 21

Pack 1 • Paper 2 • Question 21

05:36

### 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.