### Video Transcript

A pick and mix bag contains the following sweets: five cola bottles, five gummy bears, and 12 milk bottles. Alfred randomly chooses a sweet from the bag. a) Write down the probability that Alfred chooses a gummy bear.

The probability of an event occurring is given by the total number of ways that this event can occur divided by the total possible number of outcomes. Assuming that each of the sweets has an equally likely chance of being chosen, to find the probability that Alfred chooses a gummy bear, we need to find the number of ways he can choose a gummy bear and divide it by the total number of outcomes — that’s all possible choices of sweets from the bag.

There are five gummy bears. There are a total of five plus five plus 12 which is 22 sweets in the bag. This means that the probability that Alfred chooses a gummy bear must be five over 22.

His friend Michael takes two sweets from the bag. b) List all the possible combinations of sweets that Michael could have chosen.

This is called systematic listing. We need to find a system that will allow us to list all the possible combinations, without losing any. It is sensible to abbreviate to save a little bit of time. Let’s call cola bottles CB, gummy Bears GB, and milk bottles MB.

If his first sweet is a cola bottle, it makes sense that he could have also picked another cola bottle. Alternatively, he could have picked a cola bottle and a gummy bear. The final possible option if one of his sweets was a cola bottle is for the other one to be a milk bottle. That’s all the possible options had he chosen a cola bottle.

Now, imagine his first sweet was a gummy bear. We’ve already said that he could have picked a gummy bear and a cola bottle. So we won’t repeat that one. However, he could have chosen two gummy bears. And alternatively, he could have chosen a gummy bear and a milk bottle. That’s all the possible outcomes had one of his sweets been a gummy bear.

Now, let’s imagine he’s chosen a milk bottle. We’ve already said he could have chosen a cola bottle and a milk bottle or a gummy bear and a milk bottle. So we don’t need to repeat these ones. In fact, the only one we haven’t listed is the option of getting two milk bottles.

And that’s all our possible combinations: two cola bottles, a cola bottle and a gummy bear, a cola bottle and a milk bottle, two gummy bears, a gummy bear and a milk bottle, or two milk bottles.