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# Last One Standing

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Age 14 to 16

Challenge Level

- Problem
- Getting Started
- Student Solutions
- Teachers' Resources

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.