a bag contains 7 red chips and 9 blue chips. Two chips are selected randomly from the bag without replacement. What is the probability that the two chips are the same color?
If someone could just point me in the right direction for the many equations I could use I would be grateful. I'm stumped!
To solve this problem, you can use the concept of probability.
First, let's find the probability of selecting two red chips.
The probability of picking the first red chip is 7/16 (since there are 7 red chips out of a total of 16 chips).
After removing one red chip, there will be 6 red chips left out of a total of 15 chips. Therefore, the probability of picking the second red chip is 6/15.
To find the probability of both events happening (picking two red chips), we need to multiply the probabilities.
So, the probability of picking two red chips is (7/16) * (6/15).
Similarly, you can find the probability of picking two blue chips using the same method.
The probability of picking the first blue chip is 9/16, and after removing one blue chip, the probability of picking the second blue chip is 8/15.
Now, you can calculate the probability of picking two chips of the same color by adding the probability of picking two red chips and the probability of picking two blue chips.
Probability of picking two chips of the same color = (7/16) * (6/15) + (9/16) * (8/15).
Simplifying this expression will give you the final probability.
To find the probability that the two chips are the same color, you can use the concept of conditional probability.
Here's how you can approach the problem:
Step 1: Determine the total number of ways to select two chips from the bag without replacement. This can be calculated using the combination formula: nCr = n! / [(n-r)! * r!], where n is the total number of chips and r is the number of chips to be selected. In this case, n = 16 (7 red + 9 blue) and r = 2. So, the total number of ways to select 2 chips is 16C2 = 16! / (14! * 2!) = 120.
Step 2: Determine the number of favorable outcomes, which means selecting two chips of the same color. This can be done in two ways: selecting 2 red chips or 2 blue chips.
For selecting 2 red chips:
The probability of selecting the first red chip is 7/16 (since there are 7 red chips out of 16 total chips initially). After one red chip is selected, there are 6 red chips remaining out of 15 total chips. So, the probability of selecting the second red chip is 6/15. Therefore, the probability of selecting 2 red chips is (7/16) * (6/15) = 42/240.
Similarly, for selecting 2 blue chips:
The probability of selecting the first blue chip is 9/16 (since there are 9 blue chips out of 16 total chips initially). After one blue chip is selected, there are 8 blue chips remaining out of 15 total chips. So, the probability of selecting the second blue chip is 8/15. Therefore, the probability of selecting 2 blue chips is (9/16) * (8/15) = 72/240.
Step 3: Add the probabilities of selecting 2 red chips and 2 blue chips to get the total number of favorable outcomes: (42/240) + (72/240) = 114/240.
Step 4: Calculate the probability by dividing the number of favorable outcomes by the total number of possible outcomes: (114/240) / (120/240) = 114/120 = 19/20.
Therefore, the probability that the two chips selected are of the same color is 19/20 or approximately 0.95.
If you have any additional questions or need further clarification, feel free to ask!
The probability of getting two red chips = 7/16 times 6/15.
The probability of getting two blue chips = 9/16 * 8/15.
The probability of getting either one or the other is found by adding those two products.
I'll let you do the calculations.