An experiment consists of recording in order of their births, the sex composition of a four child family in which the children were born at different times.

Describe an appropriate sample space s for this experiments

I can do
GGGB, BBBG,GBGB,BGBG,BBGG,GGBB,BGGG,GBBB..... but is there a less tedious method to do this?

birthday 1, 2 choices

birthday 2, 2 choices
birthday 3 , 2 choices
birthday 4, 2 choices
2*2*2*2 = 2^4 = 16

b b b b
b b b g
b b g b
b b g g
---------
b g b b
b g b g
b g g b
b g g g
----------------
g g g g
g g g b
g g b g
etc

No, that's about all you can do.

Just your luck it was only for 4 children, which means there would be 2^4 or 16 cases.
Just go about it systematically.
Just imagine if there had been 6 kids.

Yes, there is a less tedious method to describe the sample space for this experiment.

To construct the sample space, we can use the concept of combinations. Since each child can be either a boy (B) or a girl (G), there are 2 possible outcomes for each child.

In this experiment, we are interested in the sex composition of a four-child family. The possible options are:

1. 4 girls (GGGG)
2. 3 girls and 1 boy (GGGB, GGBG, GBGG, BGGG)
3. 2 girls and 2 boys (GGBB, GBGB, BGBG, BGGB, BBGG)
4. 1 girl and 3 boys (GBBB, BGBB, BBGB, BBBG)
5. 4 boys (BBBB)

So, the sample space (s) for this experiment consists of these 14 possible outcomes: {GGGG, GGGB, GGBG, GBGG, BGGG, GGBB, GBGB, BGBG, BGGB, BBGG, GBBB, BGBB, BBGB, BBBG}.

Yes, there is a less tedious method to determine the sample space for this experiment. One approach is to use the concept of permutations.

In this case, we are interested in the sex composition of a four-child family, where the order of the births matters and the children are born at different times. Since each child can either be a boy (B) or a girl (G), there are two possible outcomes for each birth.

To determine the sample space, we can use the concept of permutations with repetition. In this case, we have 2 possibilities (boy or girl) for each of the four births, with the order of the births being important.

The formula for permutations with repetition is P(n, r) = n^r, where n is the number of possibilities for each event and r is the number of events.

In our case, n = 2 (boy or girl) and r = 4 (number of births). Applying the formula, we get:

P(2, 4) = 2^4 = 16

Therefore, the sample space for this experiment consists of 16 possible outcomes, representing all the different combinations of boys (B) and girls (G) in a four-child family where the order of births matters and the children are born at different times.