# A bag contains 15 red candies,15 yellow candies, and 15 blue candies. What is the probability of randomly choosing two candies of the same color, without replacement?

Here we need to consider few things first

1. We are not replacing
2. There are totally 45 candies in the bag
3. 15 red, 15 yellow and 15 Blue.
4. It is to find probability of picking two SAME color candies.

Here are the steps:

It is to pick two same color candies without replacement.

So, we have three options to pick three same color candies

They are two red or two yellow or two blue.

So, we can write probability as P(red, red) + P(yellow, yellow) +P(Blue, Blue)

Let’s find each separately.

P(red, red)= P(red)* p(red)

There are 45 candies in bag , out of which 15 are red.

So, P(first red)= 15/45

We are NOT replacing this red back into bag.

So, we have 14 red out of 44 candies in the bag.

P(second red)= 14/44

So, P(red, red) =15/45 *14/44

Like so far P(Yellow, Yellow) = 15/45 *14/44

Like so for P(Blue, Blue) = 15/45*14/44

Now, add them together and simplify

15/45 *14/44 + 15/45*14/44 +15/45*14/44

Simplify each fraction, 1/3*7/22 +1/3 *7/22 + 1/3*7/22

Simplify each again

7/66 +7/66+7/66

Add the like fractions

(7+7+7)/66

21/66

Common factor for both top and bottom is 3.

So, divide each by 3.

We get 7/22

Answer is 7/22

If you have any questions on any of the steps, please leave a comment! I’m more than happy to explain it in detail:)

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