Answer for the question ---Question No. 02
a) Find the sample space for the gender of the children if a family has three children. Use B for boy and G for girl. (04 Marks)
(Total 20 Marks)
视频信息
答案文本
视频字幕
We need to find the sample space for the gender of children in a family with three children. A sample space is the set of all possible outcomes of an experiment. In this case, each child can be either a boy, represented by B, or a girl, represented by G. Since we have three children, we need to list all possible combinations of these outcomes.
Let's systematically enumerate all possibilities. Each child has 2 possibilities: boy or girl. With 3 children, we have 2 to the power of 3, which equals 8 total outcomes. We can organize these into categories: one outcome with all boys, three outcomes with 2 boys and 1 girl, three outcomes with 1 boy and 2 girls, and one outcome with all girls. This tree diagram shows all possible combinations.
Here is the complete sample space. It contains eight unique outcomes: BBB, BBG, BGB, BGG, GBB, GBG, GGB, and GGG. We can verify this is correct because we have 8 total outcomes, which matches our formula of 2 to the power of 3. All outcomes are unique, and we have covered all possible combinations of boys and girls for three children.
We can analyze the sample space by categories based on the number of boys and girls. There is 1 outcome with all boys: BBB. There are 3 outcomes with 2 boys and 1 girl: BBG, BGB, and GBB. There are 3 outcomes with 1 boy and 2 girls: BGG, GBG, and GGB. And there is 1 outcome with all girls: GGG. The total is 1 plus 3 plus 3 plus 1, which equals 8. This follows the binomial coefficient pattern.
To summarize what we have learned: A sample space contains all possible outcomes of an experiment. For three children with two gender possibilities each, we get 8 total outcomes using the formula 2 to the power of 3. The complete sample space includes all combinations from BBB to GGG. Systematic enumeration ensures we miss no outcomes and avoid duplicates. This fundamental method applies to any probability experiment with discrete outcomes.