Maria has cloth in four colors!

Question

Maria has cloth in four colors (red,blue,green,yellow) and two styles (large,small) for the school logo.The flag she is making will have two different stripes with white in the middle and the logo.What is the probability that her design will have red and blue stripes and the large logo?

Answer 1

In general, the probability is defined as the fraction (number of favorable outcomes)/(total number of outcomes). So, we just need to find the values for the numerator and denominator. So, let's look at the denominator (total number of outcomes) first. The logo can either be large or small. So first let's assume it's large. Then, we need to choose two colors from the four that are available. So assuming, say, choosing green and yellow is the same as choosing yellow and green, we calculate the number of ways of choosing 2 items from a set of 4 as follows: (4!)/((2!)(2!)), where the "!" is the factorial of the number, which is defined as the product of the decreasing set of integers from the starting integer down to 1. So, 4! = 4*3*2*1 = 24. Similarly, 2! = 2*1 = 2. So, our value is simply (24)/(2*2) = 24/4 = 6. So, there are 6 possible ways of choosing two colors from four, assuming the logo is large. Similarly, there will be six ways of choosing two colors from four when the logo is assumed small. So the total number of outcomes is 6 + 6 = 12. So 12 is our denominator in the fraction described in the beginning. Now, we need to find the numerator, which is the number of favorable outcomes. So, there is only one favorable outcome: choosing the colors red and blue, and making the logo large. So, our numerator is simply 1. Therefore our probability is 1/12, or 0.0833333..., or about 8.33%.

 

Another way of doing this problem is taking the product of the probabilities for each "part" of the flag. That means we take the product of the probability of choosing red and blue with the probability of making the logo large. So, we already know that there are six total ways of choosing 2 colors from 4, and there's only one favorable outcome (choosing red and blue), so the probability of choosing red and blue is simply 1/6. For the logo, there are a total of 2 ways of choosing 1 logo from 2 possibilities (choosing one or the other), and there is only one favorable outcome (choosing large). So the probability of choosing a large logo is 1/2. Therefore, the probability of choosing red and blue AND choosing a large logo is (1/6)x(1/2)=1/12, which matches the answer we got in the first method.

Answer 2

 

here are all the combinations for the colors

red blue

red green

red yellow

blue green

blue yellow

green yellow

 

now add the style

red blue Large

red blue small

red green Large

red green small

red yellow Large

red yellow small

blue green Large

blue green small

blue yellow Large

blue yellow small

green yellow Large

green yellow small

 

the probability is 1/12

 

if you understand factorial you can get the answer by using the formula n!/r!(n-r)!

4!/2!2! * 2!/1!1!=(4*3)/2 times 2=12/2 times 2=6*2=12 combinations, one of which will be red blue Large