Probability through Lotto
This activity focuses on one aspect of the statistics strand of Mathematics in the New Zealand Curriculum – namely probability. The achievement aim for probability is for students to develop the ability to estimate probabilities and to use probabilities for prediction.
The aim of this activity is to develop our own understanding of probability so that we are better able to teach this concept in our classrooms. I do this through a lotto problem that incorporates many of the essential notions of probability. The idea that we want to discuss is probability and the chances of winning first prize in Lotto. This is a chance to talk about the basics of probability and to apply these in simple and complicated situations.
The format below will include both key questions and copymasters that you can use as a framework for the discussion.
Although I have presented this for two meetings you may want to condense it for only one. Alternatively, you may want to leave out some aspect of the overall discussion.
Main Question: What are your chances of winning the first prize in Lotto?
Allow some discussion on this. What do the staff actually think? How would they go about finding an answer if you said that they couldn’t go home until they had an answer?
This by itself is probably too big a problem to tackle. It would help if we first of all knew how to find simple probabilities and after that we would need to develop strategies to tackle more difficult probability questions. So we take this discussion in four stages plus an extension.
- What are the chances of something specific happening - what is the probability of an event?
- How do we use the information we have or can find, to find the chance of simple things happening – how do we find the probability of simple events?
- How do we find out all of the things that can happen – how do we list or calculate all possible outcomes?
- What are the chances of winning first prize in Lotto?
Extension: what is the probability of winning any Lotto prize?
Stage (i). So we’ll first try simple probabilities. To do this we need to know what things can happen and what things we would like to happen.
Key Questions 1: What is the probability of something happening? Is there a formula or definition that we can use to get started?
Discuss the replies. We could sneak up on this one but the probability of an event is a number between 0 and 1 inclusive and it is just "the number of favourable outcomes" / "the total number of outcomes".
Stage (ii). Let’s see this in action.
Key Question 2: What is the probability of getting a head when we toss a coin?
After maybe a little discussion you should realise that the number of ways the event could happen is just one. On the other hand, there are two possible outcomes, either we would get a head or we would get a tail. So, the probability of getting a head is "the number of ways to get a head" / "the number of possible outcome"s = 1/2.
Try one for yourself.
Key Question 3: What are the chances are of getting a 6 when you roll a dice?
What is it? Get the staff to give you the answer and justify it using the formula above.
OK, so the number of ways a 6 can turn up is just 1 and the number of possible outcomes is 6 (we can get 1, 2, 3, 4, 5 and 6). So, the the probability of getting a 6 when you roll a dice once = 1/6.
It is worth spending a little more time on dice.
Key Question 4: What’s the probability of getting an even number when you roll a dice once?
Work out the number of ways an even number can happen. Then work out the number of possible outcomes. What did you get? We get: The probability of getting an even number when you roll a dice once = 3/6 = 1/2.
Here there are three ways to get an even number (2, 4 and 6) but there are still 6 possible outcomes.
At this point you might like to discuss some more simple probabilities. Make up your own examples and see what answers you get. I’ll be happy to adjudicate if there are any controversies.
Stage (iii). So how do we find all possible outcomes and all possible occurrences of an event when it’s a bit more complicated than that? For instance, how do we find all possible outcomes when we are trying to calculate our chances of winning a Lotto first prize? Let’s try something easier first.
Key Question 5: What’s the probability of getting two numbers correct if we have to choose from 5 numbers?
Get the staff to think about how they could use the formula. First of all how can we get the two numbers? Well there is only one way. You will have chosen them at the start. The numbers are there and fixed and there is just one pair.
So then what is the number of possible outcomes here? Well, what we are trying to do is to pick one pair among all possible pairs. How many ways can we choose a pair of numbers from 5 given numbers?
The simplest way of approaching this is to make a list. And we can do that if we are careful. In fact it’s important to be careful so that we don’t miss anything. So we need to make a systematic list. This is what we do now.
12, 13, 14, 15
23, 24, 25
There are two things to note here. First, there is no difference between 12 and 21. The order the numbers come out of the machine is unimportant. The only important thing is the numbers themselves.
Second, in the list we made above, we worked systematically by first choosing 1 and all the numbers it could come with. Then we chose the 2 and all the numbers it could come with except 1. Then we chose the 3 and all the numbers it could come with except 1 and 2. Then we chose the 4 and all the numbers it could come with except 1, 2, and 3. The exceptions are OK as we have already listed the pairs with those numbers in.
Fine, so if we do a bit of counting we see that there are 10 possible outcomes here. So the
probability of getting one pair right, if only 5 numbers are being used is 1/10.
(If some of the staff found that hard they might like to try finding the probability of choosing two numbers out of 8. The same approach is needed. The difficulty will be the systematic list.)
The reason for looking at the question(s) immediately above and the questions below, is that they mimic the original Lotto question. This is just about choosing some set of numbers from the possible sets that could occur.
Armed with the experience above, try working out the probability of getting 3 numbers right if we had to choose from 6 numbers.
Key Question 6: What’s the probability of getting three numbers correct if we have to choose from 6 numbers?
What do the staff get this time? The difficult part here is to find the number of possible ways of getting triples from 6 numbers. And the way we do that is to make a systematic list.
123, 124, 125, 126, 134, 135, 136, 145, 146, 156
234, 235, 236, 245, 246, 256
345, 346, 356
Take note how the systematic list works here. Put 1 first and then 2. Then take all the triples that can come with 1 and 2. When that is done move on to 1 and 3, then 1 and 4 and 1 and 5. That exhausts all the possibilities for 2, and then 3 and then 4.
We get 20 triples on our list, so the probability of choosing one triple correctly from triples made up from 6 numbers is 1/20.
If you are taking this over two sessions, then this may be a good place to stop after the first meeting.
Quickly recall the main question, the definition of probability and the problems we have done so far.
Stage (iv). Clearly now we need to know how many ways we can choose 6 numbers from 40. This is because in Lotto you have to choose 6 numbers and there are 40 to choose from. If we follow on from what we have done above we would need to make a systematic list. But that list is going to be very big and would take an army to work it out in any reasonable time. At this point we have to get clever and count without counting
Let’s go back to the two number case. We want to choose a pair from all pairs that we can get from 5 numbers.
Key Question 7: How to do this without doing it?
Do the staff have any ideas? Well first we could see how many ways there are of choosing the first one of the pair. There are 5 numbers so there are 5 ways of choosing the first one of the pair. Now how many numbers are left? Clearly there are 4. So there are 4 ways of choosing the second member. Since 5 x 4 = 20, there must be 20 pairs that we can choose from 5 numbers.
But wait on. We only got 10 pairs when we made our list? What has gone wrong? In the recent count we said that we could have any of 5 numbers first. So we could start with 1, 2, 3, 4 or 5. Then we have 4 choices next. So we can get
12, 13, 14, 15 and 21, 23, 24, 25 and 31, 32, 34, 35 and 41, 42, 43, 45 and 51, 52, 53, 54.
That is 20. Ah! But we’ve counted the pair 12 twice. It comes first as 12 and then as 21. Similarly we get every other pair twice. So we need to divide this 20 by 2 to get the right answer – 10. So we should have counted (5x4)/2 .
That means that we have a quick way to count pairs.
Key Question 8: How do we count the number of pairs that we can get from 8 numbers?
Try to do this in two ways. First make a systematic list, then use a counting argument similar to the one we’ve just used. The systematic list is on the overhead slide and the other method gives (8 x 7)/2 = 28.
What about triples though?
Key Question 9: Can we count the number of triples we can get from 6 numbers ‘without counting’?
We know the answer by using a systematic list (see Key Question 6). So how do we do it ‘without counting’? Right, now there are 6 choices for the first number, 5 for the second and 4 for the third. This gives us 6 x 5 x 4 = 120. But that is six times as many as we got with our systematic list. So we must have counted the triples more than once.
How often could we have counted the triple 123? Well it could have been 123 or 132 or 213 or
231 or 312 or 321. That’s 6 ways. So the count should be (6x5x4)/6 = 20, just what we got from the systematic list.
Before we get back to Lotto, it would be nice to see where the 6 came from in the bottom of the fraction above. It turns out that there are 3 ways of choosing the first number in a triple, 2 ways of choosing the second number and 1 way of choosing the third. That gives 3 x 2 x 1 = 6. So the counting without counting should look like (6x5x4)/(3x2x1) . (What does the pairs count look like using this new idea for the denominator?)
Finally we are in a position to tackle Lotto.
Key Question 10: How many ways can you choose 6 numbers from 40?
We need to choose 6 numbers from 40. Fine, there are 40 choices for the first number, 39 for the second, 38 for the third, 37 for the fourth, 36 for the fifth and 35 for the sixth to give 40 x 39 x 38 x 37 x 36 x 35 (that’s a pretty big number!)
But there are 6 numbers so above we have counted each possibility in 6 x 5 x 4 x 3 x 2 x 1 ways. This means that the number of ways of choosing 6 numbers from 40 is
This means that the probability of winning the first prize is
1 in 3 838 380.
So that’s why I haven’t won it lately!
Key Question 11: What is the probability of becoming prime minister?
Let the staff argue about this until they have a rough idea of what they think the answer might be. This really needs some estimation. I’m not sure that we can do this very accurately. How many adults are there in the country who are eligible to be prime minister? 2.5 million? So the probability of being PM 1/(2.5 million). And this is larger than 1/3 838 380 which means that we all have a better chance of being prime minister than winning Lotto!
Stage (v). So what now? In maths there is always another problem to answer.
Key Question 12: Well, can you tell now what your chance of winning prize in Lotto is?
You’ll need to find out all the possible probabilities and add them together. Over to you. That should give you enough fuel for another staff meeting.
PS: The method we have used here of working out how to do things by first doing some smaller examples is a good strategy. It’s something that often proves to be very useful.