Math  /  Data & Statistics

QuestionSuppose a drawer contains four green socks, five red socks, and three white socks. We draw one sock from the drawer and it is equally likely that any one of the socks is drawn. Find the probabilities of the events in parts (a)-(e).
a. Find the probability that the sock is red. (Type an integer or a simplified fraction.)
b. Find the probability that the sock is green or white. (Type an integer or a simplified fraction.)
c. Find the probability that the sock is brown. (Type an integer or a simplified fraction.)
d. Find the probability that the sock is not green. (Type an integer or a simplified fraction.)
e. We reach into the drawer without looking to pull out four socks. What is the probability that we get at least two socks of the same color? (Type an integer or a simplified fraction.)

Studdy Solution

STEP 1

What is this asking? We're figuring out the chances of picking different colored socks from a drawer, including the chance of getting at least two of the same color when grabbing a handful! Watch out! Don't mix up "or" and "and" probabilities, and make sure to count the total number of socks correctly!

STEP 2

1. Probability of Red
2. Probability of Green or White
3. Probability of Brown
4. Probability of Not Green
5. Probability of at Least Two Same Color

STEP 3

Alright, let's **dive in**!
We want to find the probability of picking a red sock.
The probability of an event is the number of ways that event can happen divided by the total number of possible outcomes.

STEP 4

There are 55 **red socks**, and a **total** of 4 green+5 red+3 white=124 \text{ green} + 5 \text{ red} + 3 \text{ white} = 12 socks.

STEP 5

So, the probability of picking a red sock is 512\frac{5}{12}.
Easy peasy!

STEP 6

Now, we want the probability of picking a green *or* a white sock.
Since a sock can't be both green and white at the same time, we can just add the individual probabilities.

STEP 7

There are 44 **green socks** and 33 **white socks**.
So, there are 4+3=74 + 3 = 7 ways to pick a green or white sock.

STEP 8

The probability is 712\frac{7}{12}.
Boom!

STEP 9

Hmm, are there any brown socks?
Nope!

STEP 10

So, the probability of picking a brown sock is 012=0\frac{0}{12} = 0.
It's impossible!

STEP 11

We can find the probability of *not* picking a green sock by finding the probability of picking a red or white sock, just like we did before!
Or, we can be clever and subtract the probability of picking a green sock from one.

STEP 12

There are 1212 total socks and 44 green socks.
The probability of picking a green sock is 412\frac{4}{12}.

STEP 13

The probability of *not* picking a green sock is 1412=1212412=812=231 - \frac{4}{12} = \frac{12}{12} - \frac{4}{12} = \frac{8}{12} = \frac{2}{3}.
Fantastic!

STEP 14

Now for the grand finale!
We're pulling out four socks, and we want to know the probability of getting at least two of the same color.
It's easier to find the probability of the *opposite* event (getting all different colors) and subtract that from 11.

STEP 15

We only have three colors: green, red, and white.
If we pull out four socks, we *must* have at least two of the same color!
It's impossible to have all four socks be different colors.

STEP 16

So, the probability of getting all different colors is 00.

STEP 17

Therefore, the probability of getting at least two of the same color is 10=11 - 0 = 1.
It's a sure thing!

STEP 18

a. 512\frac{5}{12} b. 712\frac{7}{12} c. 00 d. 23\frac{2}{3} e. 11

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