Barbara opened a bag of candies and found that there were 8 red, 6 orange, 4 green, 5 yellow and 3 purple candies in the bag. If Barbara pulls out two of the candies at random, what is the probability that the two candies are different colors? Express your answer as a common fraction.
1 - the probability of two the same
so
two red = 8/26 * 7/25
two orange = 6/26 * 5/25
etc
add them up
subtract that from 1
To solve this problem, we can use the concept of combinations.
First, let's find the total number of ways Barbara can choose two candies from the bag. This can be calculated as the combination of 2 candies from a total of 26 candies (the sum of all the colors).
The formula for combinations is given by:
C(n, r) = n! / (r!(n-r)!)
Where n is the total number of items and r is the number of items chosen.
For our problem, n = 26 (total candies in the bag) and r = 2 (Barbara pulls out two candies).
C(26, 2) = 26! / (2!(26-2)!) = 26! / (2!24!) = (26 * 25) / (2 * 1) = 325
Next, let's find the total number of ways Barbara can choose two candies of the same color.
She can choose 2 red candies in C(8, 2) ways, 2 orange candies in C(6, 2) ways, 2 green candies in C(4, 2) ways, 2 yellow candies in C(5, 2) ways, and 2 purple candies in C(3, 2) ways:
C(8, 2) = 8! / (2!(8-2)!) = 8! / (2!6!) = (8 * 7) / (2 * 1) = 28
C(6, 2) = 6! / (2!(6-2)!) = 6! / (2!4!) = (6 * 5) / (2 * 1) = 15
C(4, 2) = 4! / (2!(4-2)!) = 4! / (2!2!) = (4 * 3) / (2 * 1) = 6
C(5, 2) = 5! / (2!(5-2)!) = 5! / (2!3!) = (5 * 4) / (2 * 1) = 10
C(3, 2) = 3! / (2!(3-2)!) = 3! / (2!1!) = (3 * 2) / (2 * 1) = 3
Now, let's calculate the total number of ways Barbara can choose two candies of different colors.
We can subtract the total number of ways she can choose two candies of the same color from the total number of ways she can choose two candies.
Total number of ways to choose two candies of different colors = Total number of ways to choose two candies – Total number of ways to choose two candies of the same color
= 325 - (28 + 15 + 6 + 10 + 3)
= 325 - 62
= 263
Finally, let's calculate the probability by dividing the total number of ways Barbara can choose two candies of different colors by the total number of ways she can choose two candies.
Probability = Number of ways to choose two candies of different colors / Number of ways to choose two candies
= 263 / 325
Therefore, the probability that the two candies are different colors is 263/325.
To solve this problem, we need to find the total number of possible pairs of candies that Barbara can pull out, as well as the total number of pairs that have different colors.
Step 1: Find the total number of possible pairs of candies.
The total number of candies in the bag is 8 + 6 + 4 + 5 + 3 = 26. Therefore, the number of possible pairs of candies that Barbara can pull out is given by the equation: C(26, 2) = (26!)/(2!(26-2)!) = 325.
Step 2: Find the total number of pairs with different colors.
To find this, we need to consider all the different combinations of different-colored candies Barbara can pull out.
The number of ways to choose a candy of one color is given by the sum of C(n, 1) for each color, where n is the number of candies of that color. So, the number of ways to choose one red candy is C(8, 1) = 8, one orange candy is C(6, 1) = 6, one green candy is C(4, 1) = 4, one yellow candy is C(5, 1) = 5, and one purple candy is C(3, 1) = 3 in this case.
Now, since Barbara wants to pull out two candies of different colors, we need to consider all the possible pairs of different-colored candies.
The total number of pairs Barbara can pull out is the sum of the product of the number of ways to choose one candy of one color and the number of ways to choose one candy of another color for each combination of colors. So, the total number of pairs with different colors is given by:
(8 * (6 + 4 + 5 + 3)) + (6 * (4 + 5 + 3)) + (4 * (5 + 3)) + (5 * 3)
Simplifying this expression gives:
(8 * 18) + (6 * 12) + (4 * 8) + (5 * 3) = 144 + 72 + 32 + 15 = 263.
Step 3: Calculate the probability.
Finally, we can calculate the probability that the two candies Barbara pulls out are different colors by dividing the number of pairs with different colors by the total number of possible pairs:
Probability = Number of pairs with different colors / Total number of pairs
Probability = 263 / 325
Therefore, the probability that the two candies Barbara pulls out are different colors is 263/325.