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 2^{3} or 8. In general, the number of possible combinations after n flips is 2^{n}.
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}/_{2}3 which is ^{1}/_{8}. In general, the probability of getting all heads (or all tails) after n flips is ^{1}/_{2}n.
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)
2^{r} 2^{r}
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.
First, we know the denominator is 2^{5} = 32.
To find the numerator, we need to calculate C(5,3) [or C(5,2)].
C(5,3) = 10.
The answer is ^{10}/_{32} = ^{5}/_{16}.
Ex [2] A coin is tossed 4 times. Find the probability of getting at least 2 heads.
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.
This means we will have to calculate each probability and add it together.
The probability of each one successively is: ^{C(4,2) + C(4,3) + C(4,4)}/_{2}4.
This is ^{6+4+1}/_{16} = ^{11}/_{16}.
The answer is ^{11}/_{16}.