Coin throws

This task is about the chance of heads or tails.

Question

A coin is fair if each throw is just as likely to come up heads (H) as tails (T).

a) If a fair coin is thrown 4 times, which is most likely?

- A total of 4 heads and 0 tails
- A total of 3 heads and 1 tail
- A total of 2 heads and 2 tails
- They are all equally likely.

Explain the reasons for your choice.

Teaching and learning:

The key concept in this resource is that there are more combinations of outcomes which have 2 heads and 2 tails than there are for any other outcomes. There is only one way of getting 4 heads (HHHH), four ways that 3 heads and 1 tail can occur, and six ways 2 heads and 2 tails can occur. However, each individual outcome is equally likely (e.g., TTTT is as likely as TTHH or HHTT).

See Throwing a coin which explores a very similar situation.

For more information about probability, look at the Probability concept map

Diagnostic and formative information:

Many of the following misconceptions suggest that the student is not yet fully grasping some central aspects of probability.

Incorrect or incomplete responses	Likely misconception
"A total of 2 heads and 2 tails" [Option 3] with an explanation similar to these: The chances are 50/50 so all the throws are equally likely. There's a 50% chance either way. If you flip a coin 4 times the chances are equally likely. Unless one side is heavier its equal. If the coin is fair then it should land on heads and tails an equal number of times.	Ignores all combination of events (Representative approach) Sees 2 Heads and 2 Tails as the best representation of all possible throws of a coin four times.
"A total of 2 heads and 2 tails" [Option 3] with an explanation similar to these: Because it is a 50:50 chance (50/100). 50% of 4 is 2. So 2 heads and 2 tails. There are two options, 4 divided by 2 is 2	Confuses frequency with probability (Representative approach) Thinks a probability of ½ (2/4) means that a half of the throws should be Heads. The student believes that 2 out of the 4 throws should be Heads, and that this is therefore most representative of what should happen.
"They are all equally likely" [Option 4] with an explanation similar to these: Because when throwing a coin up, the outcome can and could be anything. Because there is a chance of every option to happen, and there isn't an average for us to go off to determine a completely certain answer. There is no way to determine unless actually doing the "experiment"	Believes nothing can be predicted with probability (Outcome approach) These students believe that you cannot tell anything about probability. They would think that 2 Heads and 2 Tails in total is just as likely as 4 Heads and 0 Tails.
"They are all equally likely" [Option 4] with an explanation similar to this: Things like whether heads or tails is facing before flipping it, or the force put into the flip could affect the chance, so any of these could happen.	Probability can be influenced (Lack of independence) 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.

Based on a sample of 123 Year 10 students.

Next steps:

Further discussions with students may help. These can be linked with practical experiences involving probability.

Ignores all combinations of events (Representative approach)
This seems to conflict with the idea that the most likely total number of Heads is 2.
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). One way to get all combinations is to draw a tree diagrams.
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).

Confuses frequency with probability (Representative approach)
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 4.

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.

Believes nothing can be predicted with probability (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.

Probability can be influenced (Lack of independence)

Students could explore if they can flip a coin in a way which could lead to them predicting accurately what the outcome will be. We generally assume that throwing a coin or rolling a dice is "random", but is it? This could lead to a rich class discussion.

Further discussions with students on these and other misconceptions may help. These can be linked with practical experiences involving probability.

		Y10 (04/2016)
a)	"A total of 2 heads and 2 tails." [Option 3] Explanation - Any 1 of: There are more (six) ways that you can get a total of 2 heads and 2 tails than other outcomes; There is only one way you can get 4 heads and 0 tails, or 4 tails and 0 heads; There are four ways of getting 3 heads and 1 tail, or 1 head and 3 tails; Probability statements such as Prob(4 heads) = 1/16 etc. No student in the sample came up with a completely satisfactory explanation. None mentioned the different ways that 2 heads and 2 tails could occur.	moderate very difficult

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Coin throws

Coin throws

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