susan, joanne, and diane are triplets. susan has red, blue, green, and yellow sweaters. janne has green, red, purple, and white sweaters. diane's sweaters are red, blue, purple, and mauve. each girl has only one sweater of each color, and will pic a sweater t wear at random. what is the probability that each girl chooses a different color? what is the probability that each girl chooses the same color? what is the probability that two girls choose the same color and the third chooses a different color? what is the probability that each girl chooses a red sweater?
To calculate the probabilities, we need to consider the total number of possible outcomes and the favorable outcomes for each scenario.
1. Probability that each girl chooses a different color:
Since each girl has 4 different sweaters to choose from, the total number of possible outcomes is 4 x 4 x 4 = 64 (since each girl chooses independently).
The favorable outcomes are the cases where each girl chooses a different color. In this case, there can be no repetitions, so the first girl has 4 options, the second girl has 3 options remaining, and the third girl has 2 options remaining. Therefore, the number of favorable outcomes is 4 x 3 x 2 = 24.
The probability is given by the favorable outcomes divided by the total outcomes: 24/64 = 3/8.
2. Probability that each girl chooses the same color:
In this case, the girls must all choose one specific color. Since each girl has 4 options, the total number of outcomes is 4 x 4 x 4 = 64.
There is only one favorable outcome since they all have to choose the same color.
The probability is therefore 1/64.
3. Probability that two girls choose the same color and the third chooses a different color:
Let's consider the cases where two girls choose the same color and the third chooses a different one. Assume the two girls who choose the same color are Susan and Joanne (scenario 1), and the third girl is Diane.
Scenario 1: Susan and Joanne choose the same color (4 options), Diane chooses a different color (3 options).
The number of favorable outcomes in this scenario is 4 x 3 = 12.
Since this can happen in three different ways (Susan-Joanne, Joanne-Susan, Susan-Diane), we can multiply the number of favorable outcomes by 3.
The total number of possible outcomes is still 64 (as calculated in the previous scenarios).
Therefore, the probability that two girls choose the same color and the third chooses a different color is 3 x 12/64 = 3/16.
4. Probability that each girl chooses a red sweater:
Since each girl has 4 options, the total number of outcomes is 4 x 4 x 4 = 64.
To have each girl choose a red sweater, there is only one favorable outcome since each girl can only choose one specific color.
The probability is therefore 1/64.
To calculate the probabilities in this scenario, we need to determine the total number of possible outcomes and the number of favorable outcomes for each case.
1. Probability that each girl chooses a different color:
The total number of possible outcomes is the product of the number of sweaters each girl has: 4 * 4 * 4 = 64.
For the favorable outcomes, each girl needs to choose a different color sweater. The first girl has 4 choices, the second one has 3 choices (since one color is already taken), and the third one has 2 choices. Therefore, the number of favorable outcomes is 4 * 3 * 2 = 24.
The probability is given by (favorable outcomes) / (total outcomes) = 24 / 64 = 3 / 8.
2. Probability that each girl chooses the same color:
In this case, all three girls need to pick the same color sweater. They can choose from the common set of colors available to all of them, which is the intersection of the color sets each girl has: red.
Therefore, there is only one favorable outcome, and the probability is 1 / 64.
3. Probability that two girls choose the same color and the third chooses a different color:
For this case, two girls must choose the same color sweater while the third girl chooses a different color. We need to consider the different possibilities for which girl chooses a different color. There are three such possibilities.
To calculate the total number of favorable outcomes, we can arrange the girls in two steps: first selecting a color for the two girls (4 choices), then choosing a different color for the third girl (3 choices).
Therefore, the number of favorable outcomes is 4 * 3 = 12.
The probability is given by (favorable outcomes) / (total outcomes) = 12 / 64 = 3 / 16.
4. Probability that each girl chooses a red sweater:
Since each girl has one red sweater, the probability that each girl chooses a red sweater is equal to the probability that only one color is chosen, which is the probability from case 2 above: 1 / 64.
To summarize:
- Probability that each girl chooses a different color: 3 / 8
- Probability that each girl chooses the same color: 1 / 64
- Probability that two girls choose the same color and the third chooses a different color: 3 / 16
- Probability that each girl chooses a red sweater: 1 / 64