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Math Help - Explaining the 'coin-toss' problem...

  1. #1
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    Explaining the 'coin-toss' problem...

    I have a question that reads thus:
    "Peter tosses a coin four times and records the number X of heads. What are the chances that X=0, X=1, X=2, X=3, X=4?"

    I was trying to explain this to a class in a tutorial the other day...
    I know there's the formula method, which is probably the best thing to teach them, or the more tedious writing out of all possible outcomes and selecting the appropriate ones, but the class has little mathematical background and I was sure there was a "nicer" explanation, as it were.
    I've had a look through the numerous threads here where this question was answered, and I googled elsewhere, but the only methods I found were the two listed above.
    Should I just give them the formula and say - "this is how you do it!" without any explanation? Writing out all possible outcomes is clearly not feasible for a greater set of outcomes...

    I'm just hoping perhaps that one of the forum members may have a nicer 'explanation' than simply using the formula...
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  2. #2
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    Quote Originally Posted by Unenlightened View Post
    I have a question that reads thus:
    "Peter tosses a coin four times and records the number X of heads. What are the chances that X=0, X=1, X=2, X=3, X=4?"

    I was trying to explain this to a class in a tutorial the other day...
    I know there's the formula method, which is probably the best thing to teach them, or the more tedious writing out of all possible outcomes and selecting the appropriate ones, but the class has little mathematical background and I was sure there was a "nicer" explanation, as it were.
    I've had a look through the numerous threads here where this question was answered, and I googled elsewhere, but the only methods I found were the two listed above.
    Should I just give them the formula and say - "this is how you do it!" without any explanation? Writing out all possible outcomes is clearly not feasible for a greater set of outcomes...

    I'm just hoping perhaps that one of the forum members may have a nicer 'explanation' than simply using the formula...
    Given the background of the class, you should definitely take a visual approach and list all the possible outcomes (there are only 2^4 = 16 so I don't see what the problem is in doing this). And use different colors for heads and tails (perhaps blue for heads and red for tails).
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  3. #3
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    Right, but what happens when the exam question has 10 coin tosses?
    They can't write out all the combinations then...
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  4. #4
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    Hello Unenlightened
    Quote Originally Posted by Unenlightened View Post
    Right, but what happens when the exam question has 10 coin tosses?
    They can't write out all the combinations then...
    Have you thought about playing around with Pascal's Triangle? Lots of information on the web. I just Googled Pascal's Triangle.

    Each row represents the distribution of heads and tails for a given number of tosses. For example on row 4 (counting the top '1' as row 0):
    1\quad4\quad6\quad4\quad1
    the sum is
    1+4+6+4+1 = 16
    This sum represents the total number of different ways (2^4) in which the 4 coins can land.

    Each individual number in the row gives the number of ways of getting a particular number of heads. Thus
    1 represents the 1 way of getting 0 heads and 4 tails: TTTT

    4 represents the 4 ways of getting 1 head and 3 tails: HTTT, THTT, TTHT, TTTH

    ... and so on
    The probabilities associated with each of these numbers are, of course, the numbers divided by 16.

    By playing around with the first few rows of Pascal's triangle, then, it should be fairly easy to develop a feeling for the different combinations and probabilities when the numbers are small. It's then a simple matter of extending the triangle down to row 10 - or indeed, of developing the formula
    \binom{n}{r} = \frac{n!}{r!(n-r)!}
    and, if you're feeling ambitious, the Binomial probability formula
    p(r) = \binom{n}{r}p^r(1-p)^{n-r}
    which is particularly simple if p=\tfrac12:
    p(r)=\binom{n}{r}\left(\frac12\right)^n
    As an alternative, if computers are available, have you thought about using a random number generator in a spreadsheet to simulate the throws of the coins? Excel, for example, has an Analysis Tool Pack that is most useful. I found (before I retired!) this a very successful method of teaching the early concepts of probability. It's very easy to simulate a large number of experiments - begin with just two coins, for example, 'tossed' 1000 times, and look at the distribution. You'll get approximately 250, 500, 250, but it won't be exact, and can then lead into a discussion of experimental versus theoretical probability. I found that getting the computer to do all the hard work allows the students to develop a solid understanding of the basic concepts by handling realistically large amounts of data.

    Grandad
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  5. #5
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    Pascal's triangle is a fantastic idea, thanks!

    I'll avoid the computer bit as they have only a calculator in the exam, but thank you for the suggestion nonetheless.
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