Last one standing
Problem
Last One Standing printable sheet
Imagine a school assembly with 250 students. Everyone stands up and flips a coin. People with tails sit down. People with heads flip again.
Do you think anyone will get 6 heads in a row?
How many heads in a row do you expect the last one standing to have flipped?
Can you explain your reasoning?
Here is an animation for you to explore what happens when different sizes of school assembly gather and carry out the experiment.
Now that you have had the chance to explore, do your answers and reasoning to the questions above change at all?
How many people would you need to have in a school assembly for you to expect there to be someone still standing after ten flips?
Here are some related questions you might like to consider:
- Imagine that if you buy one ticket, the probability of winning the lottery jackpot is approximately 1 in 14 million. If there are usually two jackpot winners every week, how many tickets do you think are sold each week?
- On October 7th 2010, a woman gave birth to her third child. Her first two children were also born on October 7th, in 2005 and 2007. So all three children in the family have the same birthday. The odds of this happening were incorrectly reported in the newspapers as being 1 in 48 million. Can you work out the correct probability?
There are more than a million families in the UK with three children.
Would you expect there to be other families with three children who share a birthday?
- The television performer Derren Brown once filmed himself flipping ten heads in a row for a programme about horse racing and unlikely events. He used a fair coin, and kept filming until he got ten in a row. How long do you think it took him?
Getting Started
What proportion of the students standing would you expect to sit down after each flip of the coins?
Student Solutions
Many thanks to everyone who sent in their ideas and solutions to the team. You clearly had fun trying out the interactivity and testing your ideas. We received solutions from the Frederick Irwin Anglican School and the Learning Enrichment Studio in Australia, the Garden International School in Malaysia, Wilson's Scool in the UK, St Paul's School, the Diocesan Girls'
School in Hong Kong, Lancing College, and the Village High School.
Anthony and Mitchell, who both attend the Frederick Irwin Anglican School, explored the problem using the interactivity:
We tested it 5 times, we averaged the answers out to get 4.2, which is rounded to 4. So with chances, you could go up to 6 times without getting tails. It is possible but the odds are greater to flip it and land on tails before you flip heads 6 times.
Thank you, both. Mackenzie, from the Village High School, shared his insights on the possibility of someone flipping six heads in a row (we've added a note of our own to his solution):
If there's a half chance of getting heads you would need to half the number of students each time. For example: 250 125 62.5 (Why might it be helpful to round this number?) and onwards After 6 coin flips around 3-4 people would be left
Thank you, Mackenzie.
Ashton, from the Learning Enrichment Studio, adopted a similar approach:
You would have to halve the number of people because it is 50% chance of it landing on heads. So out of 250 there would be approximately 125 left because 50% would have most likely flipped heads. So again 63 people again would be halving because of 50% chance again so 31 people. Because halving again so. 15 people again halving. 8 from halving, 4 from halving, 2 from halving then 1 from halving. So 8 tosses for last person standing and 4 people for 6 tosses.
Vihaga and Leia, also from the Frederick Irwin Anglican School, thought carefully about the size of the school for this problem:
If there is a school of 250 people and they all flipped a coin approximately half would sit down because there are 2 sides to a coin and also a 50/50 chance. So, in a larger situation like 250 people there would be about 5-7 flips before the last person sits down. And in a smaller situation like 16 people there would be fewer flips like 3-5 before the person sat down. So, we can conclude in most situations the bigger the amount of people the more flips until the last person sits down.
Adavya and Aman, from St Paul's School, were two of the students who submitted solutions exploring the probability behind the results (we've added our own comment too):
The probability of getting 6 heads is 1 / 2^6, or 1/64. Since there are 250 people, or 250/64 or 3.90625 people will get 6 heads in a row (Why might it be helpful to round 3.90625 in this case?).
As we can see for this example, every round on average half of the people will sit down. Since 2^8=256 which is close to 250, the most likely scenario is that the winner flips 8 heads in a row.
Since 2^10 is 1024, you could expect there to be someone standing after 10 flips if there are 1024 people.
Ariel, who attends the Diocesan Girls' School, shared her thoughts about the follow-up questions:
For the first related problem, since the probability is 1 in 14 million, 14*2=28 million tickets are supposed to be sold each week (assume that everyone chooses their number randomly).
For the second related problem, the correct probability should be 1 in 133225 (1/36562), ignoring leap years. The incorrect answer is because of forgetting the event can happen on any date. And since there are more than 1 million families, there should be about 10 (about 1 million/100 thousand) or more families with three children sharing a birthday. Maybe coincidence is not that rare!
For the third related problem, the probability of getting 10 heads in a row is 1/2610=1/1024. If we assume each flip takes 2 seconds and he immediately restarts when he gets tails, it should take him about (512+256*2+128*3+...+4*8+2*9+10)*2=4054 seconds which is about 68 minutes, to get 10 heads in a row. Therefore, it should take him about an hour to film this unlikely event.
Thanks you for sharing your thought on those follow-up problems, Ariel.
Teachers' Resources
Why do this problem?
This problem offers an engaging context in which to discuss probability and uncertainty. Intuition can often let us down when working on probability; this problem has been designed to provoke discussions that challenge commonly-held misconceptions. You can read more about it in this article.
Unusual events are expected to happen to someone if the population is large enough. It's impossible to predict in advance who it will happen to, but after the event we shouldn't be surprised that it has occured. This problem provides an example of this phenomenon.
Possible approach
Before the lesson, prepare a sealed envelope with the word FIVE written on it (assuming you have a class of about 30).
There are lots of discussions that can come from this task and from watching the animation at each stage. For example:
- Why does the number of people standing halve at each stage?
- Is it possible to predict where the last person standing will be on the grid?
The animation could be used with each student choosing a point or region of the grid and seeing if they chose correctly, to capture the idea that the unexpected will happen to someone almost every time, but that it's difficult to predict who it will happen to.
There are some suggested questions at the end of the problem that could be used to explore the ideas further. Alternatively, the class could be asked to think of other examples where very unlikely events happen in very large populations.
Key questions
What proportion of the people standing do we expect to sit down on each flip of the coin?
Possible support
Encourage students to start by analysing what happens with only a very small number of people in the room, and to use the interactivity to model it.
Possible extension
Same Number! offers another opportunity to work with probabilities in order to explain unexpected events.