A basket contains 6 oranges and 4 tangerines. A sample of 3 is drawn. Find the probability that they are all oranges.

6/10 * 5/9 * 4/8 = ?

.1666

To find the probability that all 3 fruits drawn are oranges, we need to determine the total number of possible outcomes and the number of favorable outcomes.

Total number of possible outcomes:
We have a total of 10 fruits in the basket, so the total number of possible outcomes is given by the combination formula:

C(n, r) = n! / (r! * (n - r)!)
where n is the total number of items in the set (10 fruits in this case) and r is the number of items chosen (3 fruits in this case).

C(10, 3) = 10! / (3! * (10 - 3)!)
C(10, 3) = 10! / (3! * 7!)
C(10, 3) = (10 * 9 * 8) / (3 * 2 * 1)
C(10, 3) = 120

So, there are 120 possible outcomes when drawing 3 fruits from the basket.

Number of favorable outcomes:
We want to find the number of ways to choose all 3 fruits as oranges, which is 6 oranges in the basket. So, the number of favorable outcomes is given by the combination formula again:

C(n, r) = n! / (r! * (n - r)!)
where n is the total number of oranges in the set (6 oranges in this case) and r is the number of oranges chosen (3 oranges in this case).

C(6, 3) = 6! / (3! * (6 - 3)!)
C(6, 3) = 6! / (3! * 3!)
C(6, 3) = (6 * 5 * 4) / (3 * 2 * 1)
C(6, 3) = 20

So, there are 20 favorable outcomes when choosing 3 oranges from the basket.

Probability that all 3 fruits drawn are oranges:
The probability is given by the number of favorable outcomes divided by the total number of possible outcomes:

P(all oranges) = favorable outcomes / total outcomes
P(all oranges) = 20 / 120
P(all oranges) = 1 / 6
P(all oranges) ≈ 0.1667

Therefore, the probability that all 3 fruits drawn are oranges is approximately 0.1667 or 16.67%.

To find the probability that all three fruits drawn from the basket are oranges, we need to calculate the ratio of the number of ways to pick 3 oranges to the total number of ways to pick 3 fruits.

Step 1: Calculate the number of ways to pick 3 oranges from the 6 available oranges.
This can be done using the combination formula, also known as "n choose k." The formula is:

C(n, k) = n! / (k!(n-k)!)

In this case, we have 6 oranges and we want to choose 3, so the calculation is:

C(6, 3) = 6! / (3!(6-3)!) = 6! / (3!3!) = (6 * 5 * 4) / (3 * 2 * 1) = 20

Therefore, there are 20 ways to pick 3 oranges from 6.

Step 2: Calculate the total number of ways to pick 3 fruits from the 10 available fruits (6 oranges and 4 tangerines).
Again, we can use the combination formula:

C(10, 3) = 10! / (3!(10-3)!) = 10! / (3!7!) = (10 * 9 * 8) / (3 * 2 * 1) = 120

Therefore, there are 120 ways to pick 3 fruits from 10.

Step 3: Calculate the probability by dividing the number of ways to pick 3 oranges by the total number of ways to pick 3 fruits.

Probability = Number of ways to pick 3 oranges / Total number of ways to pick 3 fruits

Probability = 20 / 120 = 1/6

Therefore, the probability that all three fruits drawn are oranges is 1/6.