Throwing a coin

This task is about chance when throwing a coin.

David throws a fair coin six times. Each throw is just as likely to come up as Heads (H) or Tails (T).

Question

Which of the following sequences is the most likely to occur? (circle one answer below)

- H T T H T H
- T H T T H T
- H T H H H H
- T T T T T T
- They are all equally likely

Explain the reasons for your choice

Answers/responses:

Y8 (10/2009)

Y10 (11/2009)

E (They are all equally likely)

moderate

Any 1 of:

Each individual throw has the same probability, so each sequence of throws has the same probability (all equally likely);
One throw does not affect the result for the next throw (the throws are independent);
The throws are completely random, have no pattern, or are unpredictable;
The coin could come up H or T at each throw;
Each sequence has a probability of (1/2)⁶; or
Another set of six throws would give a different sequence.

difficult

moderate

Based on a representative sample of 156 Y8 and 144 Y10 students.

Examples of correct responses

Year 10
Responses suggesting the student is at or above Year 8

If you throw a Tail, next time you will not be more likely to throw a Head or Tail.
Each throw is equally likely to be H or T so the sequences are all equally likely because the coin could come up as either.
The side that it lands on is completely random each time so you could get any pattern or no pattern.
You have a 50% chance of throwing H or T. Each time you throw it does not effect (sic) the next time and each throw is not effected (sic) by the one before it.
Because it's just as likely that a H or a T will come up.
Because each throw is just as likely to come up Heads or Tails.
It's just random what comes up, any sequence could work.

Year 8
Responses suggesting the student is at Year 8

It’s a 50-50 chance and you can’t predict what would happen with the coins
Because you can’t control if the coin is going to land as a Head or Tail.
Because you can’t control which side comes up unless it is a double sided coin.
No-one can predict the order of a coin throw because it may change at any time.
Because you don’t know what’s going to be next.

Responses suggesting the student is at Year 7(unless a discussion with them indicates otherwise)

They are all equally likely as it is a chance thing. It could go either way.
Because you never know what it is going to land on, it is unpredictable.
I thought all of them would have a pretty good chance of getting it.

Teaching and learning:

The key concept that is being assessed is that of the independence of chance events. One coin toss has no effect on the next throw. If this is the case, then each sequence of throws is equally likely. Even many adults do not have this concept correct, and choose option A.
That A is incorrect seems to conflict with the idea that the most likely total number of heads is three. The reason for this conflict is that there are more ways in which three heads can occur than there are for any other number of heads. There is only 1 way that no heads occur (TTTTTT), 6 ways that one head and five tails can occur, 15 ways two heads and four tails can occur, and 20 ways three heads and three tails can occur.

See resource Coin throws for a resource which explores a very similar situation.

The belief that one event (e.g., the throw of a coin) can be independent of the next is a basic presupposition of probability. It can be debated whether independence, probability, and chance actually exists in the real world. Some people claim that given sufficient information that everything can be exactly predicted or specified. They are called "determinists" (see http://en.wikipedia.org/wiki/Determinism). Other people claim that the laws of probability do govern at least some physical events. These people follow what is known as "probabilism" (http://en.wikipedia.org/wiki/Probabilism). Which ever one of these two viewpoints is held, probability is a highly effective way of modelling many real life situations. Statistical analysis is based upon probability ideas.

For more information on probability, click on Probability concept map

Diagnostic and formative information:

Many of the following misconceptions suggest that the student is not yet performing at Year 7. Some misconceptions are relatively naïve compared with others, and many adults still have some of these misconceptions. Further discussions with the student may help.

Incorrect responses

Likely misconception

a)
b)

A and
"There should be 3 H and 3 T"

There is an equal number of H and T in that sequence.
Because it is a two sided coin there is only two outcomes H or T. With 6 throws it is more likely to be 3 of each.
The number of each letter is even.
It would be really hard to throw a coin 6 times and get 6 H.

Representative approach – ignores all combination of events.
Sees 3 Heads and 3 Tails as the best representation of all possible throws of a coin 6 times. Correctly sees that the most likely number of Heads is 3 but then conclude that a particular sequence of 3 Heads and 3 Tails is most likely. This ignores the fact that there are many ways students can get 3 Heads and 3 Tails.

a)
b)

A and
"H and T have an equal (50/50 etc) chance".

Because the coin is fair … 50% should be H and 50% should be T.
There is a 50% chance it could be H and a 50% chance it could be T.

Representative approach – confuses frequency with probability
Thinks a probability of ½ (3/6) means that a half of the throws should be Heads. The student believes that 3 out of the 6 throws should be Heads, and that this is therefore most representative of what should happen.

a)
b)

B, C or D
"Head (or tails) are more likely"

Because when David tosses his coin it is more likely to get Heads than Tails.
Tails has occurred many times [in the distractors]. Tails was 84% more likely to occur.
Because I counted 14 of them [T] and only 10 H.
Because I’ve tried it once.
It’s most likely that it will come out T first then H.
You land on H more often than T.

Availability approach - experience
These students may have experienced more of these occurrences in recent coin tosses. Even counting the number of H and T in the distractors can influence students.

Availability approach – belief
Students may have an underlying conviction that one particular outcome is more likely. It may be that "Heads" is their favourite (or that "3" is their favourite number).

a)
b)

E
"It is all a matter of chance"

Because it could be anything, it’s a one in a million chance.
This is all about probability so you will never know what is likely to come up.

Outcome approach
These students believe that you cannot tell anything about probability. They would think that 3 Heads and 3 Tails in total is just as likely as 6 Heads and no Tails.

a)
b)

A or B and
"The Heads and Tails are the most mixed up"

Because if it is on H it will likely go on to T then H and so on.
It is likely that he might get a mixture of T and H.
It would be really hard to throw a coin 6 times and get 6 H.

Representative – negative regency approach
Students (and many adults too) believe that mixed up sequences of Heads and Tails are far more likely to occur and thus are most representative of what should happen. They think that longer runs of Heads or Tails (such as in C and especially D) are unrepresentative. They may believe that the next throw after a long run of Heads should be a Tail, because "it’s Tails turn". In a random sequence of coin tosses, long sequences (or runs) of Heads or Tails occur more often than most people think.

"It depends on what it is to start with"

Because if you face the coin with the H on top and you flip it, it would land on T.
Because he probably flicked when it was on H.

Lack of independence - Probability can be influenced
Students believe that the next throw will be influenced by the last throw.

Representative – positive regency approach
This belief is the opposite of negative regency, seeing that the last throw is more likely to continue a sequence of Heads or Tails.
These students generally get part a) wrong.

a)
b)

E
It depends on other external factors

You might not throw it so high or so fast and that could change the outcome.
It is because how high you throw it, how hard you toss it, [and] where it lands, and the wind can also contribute.

Lack of independence - Probability can be influenced
Students believe that external factors such as how hard the coin is flipped, whether a wind is blowing, etc. will influence how the coin lands. These students generally still get part a) correct.

Next steps:

Representative approach – ignores all combinations of events.
This seems to conflict with the idea that the most likely total number of Heads is three (see the second bullet point under Teaching and learning).
Activity 1
Get these students to work with a simpler situation involving two coins of different denominations (10c and 20c for example).

Get them to list the different possible combinations. If they only list 3, get them to separate out "10c Heads and 20c Tails" as different from "10c Tails and 20c Heads", i.e. there are two ways to get 1 head and one tail (HT and TH).
Get them to throw the coins a large number of times, recording their results. They should observe that each of the four options is approximately as common, i.e. they are equally likely. However a total of 1 Head and 1 Tail are twice as likely because there are two ways of getting it (HT and TH).

Representative approach – confuses frequency with probability
Students need to distinguish between the probability of one throw being Heads (¹/₂) and the frequency of Heads, which could be as low as 0 or as high as 6.
Students may need to use the steps in Activity 1 to distinguish between the most likely number of Heads and Tails (which has several ways of occurring), and the probability of any particular sequence.

Availability approach – experience or belief
Students need many practical experiences of random events, i.e., ones driven by probability. They could predict that Heads is more likely (or maybe even how much more likely it is), do an experiment to see what happens in practice, and then explain the results. If the sample is large enough, students may start to see that their preconceptions do not necessarily hold. Computer simulations can help get large samples.

Outcome approach
Students need many experiences of random events, i.e., events driven by probability. Students who answer "It’s all just a matter of chance" may need to be asked to explain their reasoning. One effective approach is to use the Predict, Observe, Explain (POE) approach. They could be asked "What will the sum of 2 dice be?" and "Is each sum equally likely?" If students are using the outcome approach, they may well predict that each sum between 2 and 12 is equally likely "because it is all just chance". Doing an experiment and graphing their results will quickly dispel this misconception. If students do a tally chart or a dot plot of their results they will see that the data starts to form a shape (distribution) which becomes closer to a triangle the more times the dice are thrown. This shows that chance itself follows its own set of rules, but these are based upon the laws of probability.

Representative – Negative regency effect
This can be effectively addressed by a discussion. Ask the students: "A fair coin has come up Heads 6 times in a row. What is the next throw most likely to be? H? T? Both are equally likely?"
Get people in each group to give reasons for their choice. This activity shows another approach.
Activity 2
Ask students to write down a sequence of what they think is a random sequence of 20 H or T.

Get the students to count the number of runs. A run is a sequence of the same toss.
For example has run of lengths 2, 1, 1, 3, and 1
Get the students to do a dot plot of the run lengths.
Pool these over the whole class.
Get each student to spin a coin 20 times and record their results.
Plot the number of runs, and pool this over the whole class.

The number of long runs in actual throws will most probably be higher than the number of long runs in the predicted sequence. This is because people think that after a long run of Heads, it is Tails turn. This is negative regency!

Lack of independence - Probability can be influenced or positive regency effect
Students can see that the laws of probability and of independence are useful ways of modelling the real world by having extensive experiences of situations of chance, such as coin tosses, rolling dice, computer games (based on random numbers), computer simulations, etc. We need to thoroughly shake dice, or spin coins well before getting what looks like a random (chance) outcome. Shuffling packs of cards is seldom random because we do not shuffle them enough.

Comparison of Year 8 and Year 10 responses

Year 8 students were equally as successful as Year 10 students at picking that "all the sequences are equally likely".
About 30% of Year 8 students got both a) and b) correct compared with about 40% of Year 10 students
Year 8 students were much less likely to have a correct explanation than Year 10 students were. About 35% of theY8 students who got a) correct had an incorrect explanation (25 out of the 73 who chose E) compared with only about 10% of Y10 students (6 out of the 66 who chose E).
More Year 8 students selected options B C, or D in part a) than Year 10 students (about 20% and 5% respectively). These options were more likely to be selected by students with lower mean ability.
Of the students who got part a) incorrect, Year 8 students were far more likely to have relatively naïve, subjective explanations, whereas Year 10 students were more likely to give explanations based around attempts to quantify probability.
Click on Supporting evidence for resource for tables that support these statements.

Adapting resources with multiple-choice (selected response) questions
Resources with multiple-choice (or selected response) questions have a range of "distractor" answers. Having a whole class discussion (or group discussion) can lead to exploring why they are incorrect, or students can be asked why they gave this answer. This resource also relates to the Predict-observe-explain (POE) cycle.

Click on any of the links for further information about adapting multiple-choice (selected response) questions.

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Throwing a coin

Throwing a coin

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