A box contains 4 red and 5 blue marbles. Michele picks three marbles at random from this box. If Z is a random variable representing the # of blue marbles picked from the box, do the following.
a. Express the probability mass function of Z in tabular form.
b. Draw the corresponding histogram.
c. Compute the probability that Michele can pick more red marbles than blue from the box.
Cannot do tables or histograms here.
c. This is the probability of picking either 2 or 3 red marbles.
P(2) = 4/9 * 3/8 * 5/7
P(3) = 4/9 * 3/8 * 2/7
Either-or probabilities are found by adding the individual probabilities.
Can i ask why you multiplied it to 3/8 and 5/7 or 2/7?
To find the probability mass function (PMF) of the random variable Z, which represents the number of blue marbles picked, we can follow these steps:
a. Expressing the Probability Mass Function in Tabular Form:
1. Determine all possible values of Z. In this case, Z can take on values from 0 to 3 (since Michele picks three marbles).
2. Calculate the probability of picking each value of Z.
- P(Z = 0) = (4/9) * (3/8) * (2/7)
- P(Z = 1) = (5/9) * (4/8) * (4/7)
- P(Z = 2) = (5/9) * (4/8) * (3/7)
- P(Z = 3) = (5/9) * (4/8) * (2/7)
Therefore, the probability mass function of Z is as follows:
Value of Z | Probability (P(Z = z))
-------------|----------------------
0 | (4/9) * (3/8) * (2/7)
1 | (5/9) * (4/8) * (4/7)
2 | (5/9) * (4/8) * (3/7)
3 | (5/9) * (4/8) * (2/7)
b. Drawing the Corresponding Histogram:
To draw the histogram, you can create four bars, one for each value of Z (0, 1, 2, 3), with the height of each bar corresponding to its respective probability.
c. Computing the Probability of Picking More Red Marbles Than Blue:
To compute the probability that Michele picks more red marbles than blue, we need to sum the probabilities of the events where Z is less than or equal to 1 since picking more red marbles implies Z ≤ 1.
P(Z ≤ 1) = P(Z = 0) + P(Z = 1) = (4/9) * (3/8) * (2/7) + (5/9) * (4/8) * (4/7)
Note that this includes the case when no blue marble is picked (Z = 0) and when exactly one blue marble is picked (Z = 1).