A box contains four white, three red, and two black marbles. Three marbles are chosen at random.
a. What is the probability that all three are the same color?
b. What is the probability that all three are different colors?
a. If the events are independent, the probability of both/all events occurring is determined by multiplying the probabilities of the individual events.
P(white) = 4/9 * 3/8 * 2/7 = ?
P(red) = 3/9 * 2/8 * 1/7 = ?
P(black) = 0, since there are only 2 blacks
Either-or probabilities are found by adding the individual probabilities.
Add the above probabilities.
b. 4/9 * 3/8 * 2/7 = ?
They are not independent
To find the probability of each scenario, we need to calculate the number of favorable outcomes (marbles of the same or different colors) and divide it by the total number of possible outcomes.
Let's start with part a) - the probability that all three marbles are the same color.
Step 1: Calculate the total number of possible outcomes.
The total number of marbles in the box is 4 white + 3 red + 2 black = 9 marbles.
When we choose 3 marbles at random, we have a total of 9C3 = (9! / (3! * (9-3)!)) = 84 possible outcomes.
Step 2: Calculate the number of favorable outcomes.
To have all three marbles of the same color, we can either choose 3 white marbles, 3 red marbles, or 3 black marbles.
- The number of ways to choose 3 white marbles is 4C3 = 4.
- The number of ways to choose 3 red marbles is 3C3 = 1.
- The number of ways to choose 3 black marbles is 2C3, which is not possible as there are only 2 black marbles.
So, the number of favorable outcomes is 4 (3 white marbles + 1 red marble).
Step 3: Calculate the probability.
The probability of all three marbles being the same color is given by the number of favorable outcomes divided by the total number of outcomes.
Therefore, the probability is 4/84 or simplified as 1/21.
Now, let's move to part b) - the probability that all three marbles are different colors.
Step 1: Calculate the total number of possible outcomes (same as in part a).
The total number of possible outcomes is 84.
Step 2: Calculate the number of favorable outcomes.
To have three different color marbles, we need to choose 1 white, 1 red, and 1 black marble.
- The number of ways to choose 1 white marble is 4C1 = 4.
- The number of ways to choose 1 red marble is 3C1 = 3.
- The number of ways to choose 1 black marble is 2C1 = 2.
So, the number of favorable outcomes is 4 * 3 * 2 = 24.
Step 3: Calculate the probability.
The probability of all three marbles being different colors is given by the number of favorable outcomes divided by the total number of outcomes.
Therefore, the probability is 24/84, which simplifies to 2/7.
So, the probabilities are:
a) The probability that all three marbles are the same color is 1/21.
b) The probability that all three marbles are different colors is 2/7.