In a certain college, 55% of the students are women. Suppose, we take a sample of two students. Use a probability tree to find the probability that at least one of the two students is a woman.
To find the probability that at least one of the two students is a woman, we can use a probability tree. Here's how to construct the tree diagram:
Step 1: Divide the first branch into two possibilities: one for a woman and one for a man. Since 55% of students are women, we assign a probability of 0.55 to the woman branch and 0.45 to the man branch.
Step 2: Repeat step 1 for the second student, using the same probabilities.
Now, let's construct the probability tree:
```
Woman (0.55)
________|________
| |
Woman (0.55) Man (0.45)
| |
________ |
| | |
Woman Man Man (0.45)
(0.55) (0.45) |
|
________
| |
Woman Man
(0.55) (0.45)
```
Now, we can calculate the probability of each outcome and find the probability that at least one of the two students is a woman by adding up the probabilities of the favorable outcomes.
The favorable outcomes are:
1. The first student is a woman and the second student is a man.
2. The first student is a man and the second student is a woman.
3. Both the first and second students are women.
The probabilities for these outcomes are:
1. Woman * Man = 0.55 * 0.45 = 0.2475
2. Man * Woman = 0.45 * 0.55 = 0.2475
3. Woman * Woman = 0.55 * 0.55 = 0.3025
Adding up these probabilities, we get:
0.2475 + 0.2475 + 0.3025 = 0.7975
Therefore, the probability that at least one of the two students is a woman is 0.7975 or 79.75%.