A bag contains 5 red chips and 8 blue chips. Two chips are selected randomly without replacement from the bag. What is the probability that the two chips are of the same color? If needed, round your answer to two decimal places.
Prob(2 red) = (5/13)(4/12) = ..
Prob(2 blue) = (8/13)(7/12) = ...
You want one OR the other, so add them up
Oh, this sounds like one colorful bag of chips! Let's calculate the probability of getting two chips of the same color.
First, let's find the probability of selecting two red chips. Since we're drawing without replacement, after selecting the first chip, there will be 4 red chips left out of a total of 12 chips. So the probability of selecting another red chip is 4/12, which can be simplified to 1/3.
Next, let's find the probability of selecting two blue chips. Similarly, after selecting the first chip, there will be 7 blue chips left out of a total of 12 chips. So the probability of selecting another blue chip is 7/12.
To find the total probability, we add the probability of selecting two red chips and the probability of selecting two blue chips:
1/3 + 7/12 = 4/12 + 7/12 = 11/12 ≈ 0.92
So, the probability of selecting two chips of the same color is approximately 0.92.
To find the probability that the two chips are of the same color, we need to consider two cases: either both chips are red, or both chips are blue.
First, let's calculate the probability that both chips are red:
- There are 5 red chips out of a total of 13 chips in the bag (since one chip is already drawn and not replaced).
- So the probability of drawing a red chip first is 5/13.
- After drawing a red chip, there are 4 red chips left out of 12 chips in the bag (since one red chip has already been drawn).
- Therefore, the probability of drawing a red chip second, given that a red chip was drawn first, is 4/12 or 1/3.
- Multiplying the two probabilities together, we get (5/13) * (1/3) = 5/39.
Next, let's calculate the probability that both chips are blue:
- There are 8 blue chips out of a total of 13 chips in the bag.
- So the probability of drawing a blue chip first is 8/13.
- After drawing a blue chip, there are 7 blue chips left out of 12 chips in the bag.
- Therefore, the probability of drawing a blue chip second, given that a blue chip was drawn first, is 7/12.
- Multiplying the two probabilities together, we get (8/13) * (7/12) = 56/156.
Now, we can add the probabilities of both cases to get the final probability:
(5/39) + (56/156) = 0.2449 + 0.3589 = 0.6038.
Rounding to two decimal places, the probability that the two chips are of the same color is approximately 0.60 or 60%.
To find the probability that the two chips are of the same color, we can use the concept of combinations.
First, let's calculate the total number of ways to choose two chips from the bag. Since we are selecting without replacement, the total number of ways to choose two chips is:
Total ways = (number of chips in the bag) choose (number of chips to be chosen)
= (5 + 8) choose 2
= 13 choose 2
= 13! / (2! * (13-2)!)
= 13! / (2! * 11!)
= (13 * 12) / (2 * 1)
= 78
Next, let's calculate the number of ways to choose two red chips from the bag. Since there are 5 red chips in the bag, the number of ways to choose two red chips is:
Ways to choose two red chips = (number of red chips in the bag) choose (number of red chips to be chosen)
= 5 choose 2
= 5! / (2! * (5-2)!)
= 5! / (2! * 3!)
= (5 * 4) / (2 * 1)
= 10
Similarly, the number of ways to choose two blue chips from the bag is:
Ways to choose two blue chips = (number of blue chips in the bag) choose (number of blue chips to be chosen)
= 8 choose 2
= 8! / (2! * (8-2)!)
= 8! / (2! * 6!)
= (8 * 7) / (2 * 1)
= 28
Finally, the probability that the two chips are of the same color is:
Probability = (number of ways to choose two chips of the same color) / (total number of ways to choose two chips)
= (ways to choose two red chips + ways to choose two blue chips) / total ways
= (10 + 28) / 78
= 38 / 78
≈ 0.49 (rounded to two decimal places)
Therefore, the probability that the two chips selected are of the same color is approximately 0.49.