Pascal’s Triangle
Pascal’s triangle can be used instead of tree diagrams to see the various ways heads(yes) and tails(no) are distributed .
If X’s are the heads and Y’s are the tails, the outcome for tossing 1 coin is
the possibility: 1H + 1T
Toss 2 coins: (H + T)(H + T) = 1H2 +2HT + T2
Toss 3 coins: (H + T)( H + T)(H + T) = 1H3 + 3H2T + 3HT2 + T3
Toss 4 coins: (H + T)( H + T)(H + T)(H + T) = 1H4 + 4H3T + 6H2T2 + 4HT3 +1T4
Toss 5 coins: (H + T)( H + T)(H + T)(H + T) )(H + T) = 1H5 + 5H4T + 10H3T2 + 10H2T3 +5HT4 + 1 T5
If we arrange the coefficients of the binomial expansion, a triangle is formed. The triangle begins with 1 for the coin before it is tossed.
I
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
etc.
Students can see;
that the second number in each row tells them how many coins are tossed.
Each position in the row represents the theoretical possible number of heads.
The sum of all the numbers in a row gives the total number of outcomes that can occur.
The triangle can be continued by beginning with a 1 and then summing the pairs as you move left to right and the second number in a pair becomes the first number in the next pair.
Example: In the 5th row; the second number is 4, therefore 4 coins
are being tossed.
1 4 6 4 1 tells us that there is 1 possibility of 0 heads
4 possibilities of 1 head
6 possibilities of 2 heads
4 possibilities of 3 heads
1 possibility of 4 heads
The sum of these numbers is 16 which is the total # of outcomes.
The probability ratios are the possibility #’s divided by 16.