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Probability With Coins:

A.  Coins are a good beginning for learning probability since you are faced with only 2 choices: heads or tails.

B.  Below is a diagram of a coin after it is tossed 3 times.  This will be used as an example to explain how to use probability in relation to coins:

C.  There are several things that we can gather from looking at this diagram.

1.  The number of possible combinations after 3 flips is 23 or 8.  In general, the number of possible combinations after n flips is 2n.

2.  The probability of getting heads or tails on each successive flip is 1/2.

3.  The probability of getting all heads (or all tails) after 3 flips is 1/23 which is 1/8.  In general, the probability of getting all heads (or all tails) after n flips is 1/2n.

4.  A more complicated concept is how to determine the probability of getting 2 heads and 1 tails (or 2 tails and 1 heads).  From counting we know the answer is 3/8, but what if we had more than 3 flips?  In general, we can calculate the probability of getting x heads and y tails after r flips by using the following formula:

P = C(r,x)  or  C(r,y)

     2r            2r

D.  Let's look at some examples.  You will need to be familiar with combinations.

Ex [1]  A coin is flipped 5 times.  Find the probability of getting 3 heads and 2 tails.

  1. First, we know the denominator is 25 = 32.

  2. To find the numerator, we need to calculate C(5,3) [or C(5,2)].

  3. C(5,3) = 10.

  4. The answer is 10/32 = 5/16.

Ex [2]  A coin is tossed 4 times.  Find the probability of getting at least 2 heads.

  1. This example is a little harder as we can have 2 heads and 2 tails, 3 heads and 1 tail, or 4 heads and no tails.

  2. This means we will have to calculate each probability and add it together.

  3. The probability of each one successively is:  C(4,2) + C(4,3) + C(4,4)/24.

  4. This is 6+4+1/16 = 11/16.

  5. The answer is 11/16.


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