Parallelogram 36 Level 1 9 May 2024Life saving probabilities

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Noun: Parallelogram Pronunciation: /ˌparəˈlɛləɡram/

  1. a portmanteau word combining parallel and telegram. A message sent each week by the Parallel Project to bright young mathematicians.
  • Tackle each Parallelogram in one go. Don’t get distracted.
  • Finish by midnight on Sunday if your whole class is doing parallelograms.
  • Your score & answer sheet will appear immediately after you hit SUBMIT.
  • Don’t worry if you score less than 50%, because it means you will learn something new when you check the solutions.

1. The Poison Frog Riddle

We all know that maths is useful in everyday life, but it can also be valuable in less familiar situations.

Should you ever find yourself standing in a clearing after being poisoned by a frog and trying to decide which way to go, maths might come to your rescue!

Watch this video - feel free to pause and ponder the riddle, but don’t spend too long and be guided by your intuition.

The answer may surprise you!

(If you have problems watching the video, right click to open it in a new window)

Most people (possibly also you) decide that the direction doesn’t matter, because both the clearing and the tree stump give you a 50% chance of survival.

Yet somehow the chances of surviving at the clearing is more than 50% - let’s see why.

(Although you are - hopefully - unlikely to find yourself in a poison frog situation, probability calculations just like this are often used by doctors and medical researchers to decide the best course of action. A good understanding of how probabilities work can literally save lives!)

1 mark

1.1 With two frogs, there are four combinations of male and female, listed below.

After the additional information of hearing the male croak, which combination is eliminated, as it is no longer possible?

  • Female, Female
  • Female, Male
  • Male, Female
  • Male, Male
  • (Not answered)
3 marks

1.2 Now that the number of possible outcomes (the ‘sample space’) is reduced to 3, what are the chances of survival?

  • 13
  • 23
  • Certain
  • (Not answered)

Two of the possible three combinations contain the female antidote frog, so the chances of survival are 23.

There is a 13 chance I will be unlucky and both frogs will be male.

Knowing that one of the two frogs is male tells us, very subtly, that the other frog is more likely than not to be female.

This challenges our intuition that the probability is 12.

Like many maths problems, it will slowly make sense the more you think about it.

2 marks

1.3 If I flip a coin three times and note the result, how many combinations are possible? (Order is important!)

  • 2
  • 4
  • 6
  • 8
  • (Not answered)

As an ordered list I could have:

  1. TTT
  2. TTH
  3. THT
  4. HTT
  5. HHT
  6. HTH
  7. THH
  8. HHH

You could also consider that every coin flip has 2 options, and we do that three times, for a total of 2 × 2 × 2 = 8 outcomes.

2 marks

1.4 When flipping three coins, what is the probability that at least two of them will match?

  • 50%
  • 67%
  • 75%
  • 100%
  • (Not answered)
Show Hint (–1 mark)
–1 mark

There are eight possibilities:

  1. TTT
  2. TTH
  3. THT
  4. HTT
  5. HHT
  6. HTH
  7. THH
  8. HHH


  • all three will be the same, in which case at least two of them match
  • two will match and one will be different

There is no case where all three are different as there are only two options!

2. Some puzzles

2 marks

2.1 A shepherd had 17 sheep. However, all except 9 ran away.

How many does the shepherd have left?

  • 0
  • 8
  • 9
  • 17
  • 26
  • (Not answered)

The question tells us that all of the sheep EXCEPT 9 of them have run away. This means that 9 of them didn't run away - so the shepherd has 9 sheep left.

This is an easy question to make a mistake on, so it is a good example of why it is important to always read the wording of questions very carefully!

2 marks

2.2 I am thinking of two numbers.

When I add them I get 14.

When I take one from the other, I get 6.

What is the larger number?

  • 6
  • 8
  • 10
  • 14
  • 20
  • (Not answered)

Before you hit the SUBMIT button, here are some quick reminders:

  • You will receive your score immediately, and collect your reward points.
  • You might earn a new badge... if not, then maybe next week.
  • Make sure you go through the solution sheet – it is massively important.
  • A score of less than 50% is ok – it means you can learn lots from your mistakes.
  • The next Parallelogram is next week, at 3pm on Thursday.
  • Finally, if you missed any earlier Parallelograms, make sure you go back and complete them. You can still earn reward points and badges by completing missed Parallelograms.

Cheerio, Simon.