Four blue socks, four white socks, and four gray socks
are mixed in a drawer. You pull out two socks, one at a
time, without looking.
a. Draw a tree diagram along with the possible outcomes
and the probabilities of each branch.
b. What is the probability of getting a pair of socks of
the same color?
c. What is the probability of getting two gray socks?
d. Suppose that, instead of pulling out two socks, you
pull out four socks. What is the probability now of
getting two socks of the same color?
A drawer contains 4 red socks, 3 white socks, and 3 blue socks. Without looking, you select a sock at random, replace it, and select a second sock at random. What is the probability that the first sock is blue and the second sock is red
a. To draw a tree diagram, we can start by representing the possible outcomes for the first sock as branches, followed by the possible outcomes for the second sock branching out from each first sock branch. In this case, we have three colors: blue, white, and gray, and we will label the branches accordingly. The probabilities of each branch will be calculated as we go along.
Here is a tree diagram to represent the possible outcomes and probabilities:
B
|
-------------
| |
W G
| |
------------- -------------
| | | |
W G B W
| | | |
G B W B
| | | |
B W G G
The probability of each branch can be calculated by dividing the number of socks of that color by the total number of socks. Since we have four socks of each color, the probability of pulling out a blue sock is 4/12 = 1/3, the probability of pulling out a white sock is also 4/12 = 1/3, and the probability of pulling out a gray sock is 4/12 = 1/3.
b. To calculate the probability of getting a pair of socks of the same color, we need to add up the probabilities of all the branches where the two socks have the same color. Looking at the tree diagram, we can see that there are 4 branches where both socks are blue, 4 branches where both socks are white, and 4 branches where both socks are gray. Therefore, the total probability is (4/12) * (4/12) + (4/12) * (4/12) + (4/12) * (4/12) = (16/144) + (16/144) + (16/144) = 48/144 = 1/3.
c. The probability of getting two gray socks can be calculated by multiplying the probabilities of pulling out a gray sock on the first draw and a gray sock on the second draw. From the tree diagram, we can see that there are 4 branches where the first sock is gray and 3 branches where the second sock is gray after taking out a gray sock on the first draw. Therefore, the probability is (4/12) * (3/11) = 12/132 = 1/11.
d. If we are now pulling out four socks, instead of just two, the probability of getting two socks of the same color would be calculated differently. Let's consider the scenario where we want to get two blue socks.
To calculate the probability, we need to count the number of ways we can choose two blue socks from the four blue socks available. This is equivalent to selecting two socks from a set of four, which can be calculated using combinations. The number of ways to choose two blue socks from four is given by the combination formula: C(4, 2) = 4! / (2! * (4-2)!) = 6.
The total number of ways to choose any two socks out of the twelve available is given by C(12, 2) = 12! / (2! * (12-2)!) = 66.
Therefore, the probability of getting two blue socks out of four socks drawn is 6/66 = 1/11.
You can apply the same logic to calculate the probability of getting two socks of the same color for the other colors as well.
3/11
a. Cannot draw diagram here.
Probability of each color = 4/12 = 1/3
b. First sock = 4/12, second sock = 3/11. Probability of events all occurring is found by multiplying individual probabilities. However, you want this for either blue, white or gray. Either-or probabilities found by adding individual probabilities.
c. First step in b above.
d. With my lack of time to respond, I'll let you do this one.