A box contains 4 red marbles and 5 blue marbles. Michelle picks 3 marbles at a random. If Z is the random variable, representing the number of blue marbles picked from the box. Do the ff:
A. Express the probability function in tabular and formula form.
B. Complete the probability that Michelle can pick red marbles than blue marbles from the box.
5/9 blue
9 possibilities. cardinality is 8
A. To express the probability function in tabular form and formula form, we need to determine all the possible values Z can take and calculate the corresponding probabilities.
The possible values for Z, representing the number of blue marbles picked, are 0, 1, 2, and 3. Let's calculate the probabilities for each of these values:
1. When Z=0 (no blue marbles picked):
The probability of picking 0 blue marbles can be calculated by choosing 3 red marbles out of the 4 available red marbles and dividing it by the total number of ways to choose any 3 marbles from the box:
Probability (Z=0) = (number of ways to choose 3 red marbles) / (total number of ways to choose 3 marbles)
= (4C0 * 5C3) / (9C3)
2. When Z=1 (one blue marble picked):
The probability of picking 1 blue marble can be calculated by choosing 1 blue marble out of the 5 available blue marbles and choosing 2 red marbles out of the 4 available red marbles. Then, we divide it by the total number of ways to choose 3 marbles from the box:
Probability (Z=1) = (number of ways to choose 1 blue marble and 2 red marbles) / (total number of ways to choose 3 marbles)
= (5C1 * 4C2) / (9C3)
3. When Z=2 (two blue marbles picked):
The probability of picking 2 blue marbles can be calculated by choosing 2 blue marbles out of the 5 available blue marbles and choosing 1 red marble out of the 4 available red marbles. Then, we divide it by the total number of ways to choose 3 marbles from the box:
Probability (Z=2) = (number of ways to choose 2 blue marbles and 1 red marble) / (total number of ways to choose 3 marbles)
= (5C2 * 4C1) / (9C3)
4. When Z=3 (three blue marbles picked):
The probability of picking 3 blue marbles can be calculated by choosing 3 blue marbles out of the 5 available blue marbles and dividing it by the total number of ways to choose 3 marbles from the box:
Probability (Z=3) = (number of ways to choose 3 blue marbles) / (total number of ways to choose 3 marbles)
= (5C3 * 4C0) / (9C3)
Now, let's express the probability function in tabular form:
| Z | Probability (Z) |
|-------|----------------|
| 0 | Calculated |
| 1 | Calculated |
| 2 | Calculated |
| 3 | Calculated |
B. To complete the probability that Michelle can pick more red marbles than blue marbles from the box, we need to calculate the probabilities for Z=0 and Z=1, and then sum them up. This represents the probability of picking either 0 or 1 blue marbles:
Prob(Z=0) + Prob(Z=1) = (4C0 * 5C3) / (9C3) + (5C1 * 4C2) / (9C3)
Calculating this expression will give you the complete probability that Michelle can pick more red marbles than blue marbles from the box.