A couple plan to have exactly three children.
(a) Construct a tree diagram and list the sample space.
(b) Find the probability that the family has at least two girls.
hard to do a tree diagram here, but the sample space would be
BBB
BBG
BGB
BGG
.
.
.
GGG
"at least two" girls means either all girls or 2 girls and a boy.
count those cases and put over 8
Thanks Reiny, you are absolutly right.
(a) To construct a tree diagram for this problem, we can consider the gender of each child in the couple's family.
Starting with the first child, there are two possibilities: boy or girl. For the second child, there are two possibilities again: boy or girl. The same goes for the third child. Therefore, the tree diagram branches out with two possibilities at each level.
The tree diagram for this scenario would look like this:
B G
/ \ / \
B G B G
/ \ / \ / \ / \
B G B G B G B G
Each branch represents a different combination of genders for the three children.
(b) To find the probability that the family has at least two girls, we need to count all the outcomes where there are at least two girls and divide it by the total number of possible outcomes.
From the tree diagram, we can see that there are 4 outcomes where there are at least two girls: GGB, GBG, BGG, and GGG.
The total number of possible outcomes is 2 * 2 * 2 = 8, as each child has two gender possibilities.
Therefore, the probability that the family has at least two girls is 4/8, which simplifies to 1/2 or 0.5.
To construct a tree diagram and list the sample space for a couple planning to have exactly three children, we can start with the first child and branch out to the possibilities for the genders of the second and third child.
(a) Tree diagram and sample space:
- First Child: Boy (B), Girl (G)
- Second Child: B, G
- Third child: B, G
This results in a total of 2 x 2 x 2 = 8 possible outcomes in the sample space:
1) BBB
2) BBG
3) BGB
4) BGG
5) GBB
6) GBG
7) GGB
8) GGG
(b) To find the probability that the family has at least two girls, we need to determine the number of favorable outcomes and divide it by the total number of possible outcomes.
Favorable outcomes: GGB, GGG (2 outcomes)
Total possible outcomes: 8 (as determined from the sample space)
Therefore, the probability that the family has at least two girls is 2/8 = 1/4 or 0.25.