Suppose you conduct a study and find that the probability of having a baby boy is 60%. Now suppose three of your relatives are going to have babies.
a) Build a tree diagram showing all the conditional probabilities and joint probabilities associated with the sex of the three new babies.
b) What rule of probability are you using to obtain the joint probabilities and why? In order to answer the “why” part of the question you must tell me why the rule of probability you chose to use applies in this case.
c) What is the probability of having one boy and two girls in the three births?
Your question is incomplete. Please try again without using cut-and-paste.
a) To build a tree diagram showing all the conditional probabilities and joint probabilities associated with the sex of the three new babies, follow these steps:
1. Start by drawing a branch for the first birth. Label one branch "Boy" and the other branch "Girl."
2. From each branch of the first birth, draw two branches for the second birth labeled "Boy" and "Girl."
3. From each branch of the second birth, draw two branches for the third birth labeled "Boy" and "Girl."
The resulting tree diagram would have a total of eight terminal branches representing all possible combinations of male and female babies.
```
Boy Girl
/ \ / \
Boy Girl Boy Girl
/ \ / \ / \ / \
Boy Girl Boy Girl Boy Girl Boy Girl
```
b) In order to obtain the joint probabilities, we use the multiplication rule of probability. The multiplication rule states that the probability of two (or more) independent events occurring together is the product of their individual probabilities.
In this case, each birth can be considered as an independent event, and the probability of having a boy or a girl is consistent across each birth.
When we construct the tree diagram, we can assign a probability of 0.6 (or 60%) to the branch representing a boy and 0.4 (or 40%) to the branch representing a girl at each birth. By multiplying the probabilities along each path from the root to the terminal branches, we find the joint probabilities associated with each combination of the sex of the three babies.
c) To find the probability of having one boy and two girls in the three births, we need to identify the possible combinations that meet this criterion.
From the tree diagram, there are three possible combinations: (Boy, Girl, Girl), (Girl, Boy, Girl), and (Girl, Girl, Boy).
Since each birth is an independent event, we can simply multiply the probabilities along the corresponding branches. The probability of having a boy and two girls is:
Probability of (Boy, Girl, Girl):
0.6 (probability of a boy) * 0.4 (probability of a girl) * 0.4 (probability of a girl) = 0.096 or 9.6%
Probability of (Girl, Boy, Girl):
0.4 * 0.6 * 0.4 = 0.096 or 9.6%
Probability of (Girl, Girl, Boy):
0.4 * 0.4 * 0.6 = 0.096 or 9.6%
Therefore, the probability of having one boy and two girls in the three births is 9.6%.