It is not difficult to see the similarities between a coin toss and the chances of having either a boy or a girl because its simply one or the other. Heads or tails; boy or girl. What happens when you compare the probability of 6 coins being tossed, and six children being born in certain combinations.
Instead of guessing all of the possible combinations, both of these potential probabilities can be predicted with a little help from Pascals Triangle.
Using Pascal’s Triangle you can now fill in all of the probabilities. First,i will start with predicting 3 offspring so you will have some definite evidence that this works. You will be able to easily see how Pascal’s Triangle relates to predicting the combinations. If you don’t understand the equation at first continue to the examples and the equation should become more clear.
There are 3 steps I use to solve a probability problem using Pascal’s Triangle:
Step 1. Order the ratios and find row on Pascal’s Triangle
Step 2. Determine the X and n
X = the probability the combination will occur. Since there is a 1/2 chance of being a boy or girl we can say:
- if there are 2 children the probability that any occurrence of boy/girl is (1/2)(1/2)= 1/4
- if there are 3 children the probability that any occurrence of boy/girl is (1/2)(1/2)(1/2)= 1/8
- if there are 4 children the probability that any occurrence of boy/girl is (1/2)(1/2)(1/2)(1/2)= 1/16
n= The Pascal number that corresponds to the ratio you are looking at.
Step 3. Plug values into the equation: n*X
This may still seem a little confusing so i will give you an example. If you want to know the probability that a couple with 3 kids has 2 boys and 1 girl. You just follow the steps above:
Step 1. Order the ratios and find corresponding row on pascals triangle.
|3 boys:0 girls||1||Boy, Boy, Boy
|2 boys:1 girl||3||B B G, G B B , B G B
|1 boy:2 girls||3||G G B , B G G , G B G
|0 boys: 3 girls||1||Girl, Girl, Girl
note: the Pascal number is coming from row 3 of Pascal’s Triangle. If there were 4 children then t would come from row 4 etc…
By making this table you can see the ordered ratios next to the corresponding row for Pascal’s Triangle for every possible combination. The only thing left is to find the part of the table you will need to solve this particular problem( 2 boys and 1 girl):
ratios: 3:0, 2:1, 1:2, 0:3 — pascals row 3(for 3 children): 1, 3, 3, 1
I added the calculations in parenthesis because this is the long way of figuring out he probabilities. Using pascals triangle is the the shortcut. 1:3:3:1 corresponds to 1/8, 3/8,3/8, 1/8.
2. Determine the X and n (for 3 children)
(1/2)(1/2)(1/2)= (1/2)^3= 1/8= X.
n =3(Pascal’s number from step 1) and number of different combinations possible)
3. Plug values into the equation: n*X
Why use Pascal’s Triangle if we could just make a chart every time?… The fun stuff! Lets say a family is planning on having six children. What is the probability that they will have 3 girls and 3 boys?
Step 1. Ratios and Pascals
6:0, 5:1, 4:2, 3:3, 2:4, 1:5, 0:6. Row 6 of Pascal’s: 1, 6,15, 20, 15, 6, 1
So there are 20 different combinations with six children to get 3 boys and 3 girls. They could be BGBGBG, BBGGBBGG,….and there are 18 more possibilities. Chances are you will not be able to guess exactly those 20 possible combinations without a considerable amount of time and effort.
Step 2. Determine the X and n (6 children)
Step 3. use the equation n*X:
20*(1/64)= 20/64 or 5/16
…If you wanted to find any other combination simply change the n.
for 4 girls : 2 boy n= 15; 15(1/64)= 15/64
note: I know i haven’t posted anything in a while, but I am working on it. I’m really busy and I will try my best to post more helpful articles in the future.