A box contains 14 yellow, 6 green, 10 purple, and 12 red candies jumbled together. What is the expected number of red candies among 5 candies poured from the box?

To find the expected number of red candies among 5 candies poured from the box, we need to calculate the probability of getting different numbers of red candies and then multiply each probability by the corresponding number of red candies.

In this case, let's first calculate the probability of getting each number of red candies when pouring out 5 candies.

We have a total of 14 + 6 + 10 + 12 = 42 candies in the box.

To find the probability of getting x red candies in 5 candies, we can use combinations (nCr) formula:

P(x red candies) = (nCr) * (14/42)^x * (1 - 14/42)^(5 - x)

Where nCr represents the combinations formula, (14/42)^x is the probability of choosing x red candies, and (1 - 14/42)^(5 - x) is the probability of not choosing any more red candies.

Let's calculate the probabilities for each number of red candies:

For x = 0:
P(0 red candies) = (5C0) * (14/42)^0 * (1 - 14/42)^(5 - 0)
= 1 * 1 * (1 - 1/3)^5
= 1 * 1 * (2/3)^5
= 1 * 1 * 32/243
= 32/243

For x = 1:
P(1 red candy) = (5C1) * (14/42)^1 * (1 - 14/42)^(5 - 1)
= 5 * 14/42 * (2/3)^4
= 70/81

For x = 2:
P(2 red candies) = (5C2) * (14/42)^2 * (1 - 14/42)^(5 - 2)
= 10 * (14/42)^2 * (2/3)^3
= 280/243

For x = 3:
P(3 red candies) = (5C3) * (14/42)^3 * (1 - 14/42)^(5 - 3)
= 10 * (14/42)^3 * (2/3)^2
= 320/243

For x = 4:
P(4 red candies) = (5C4) * (14/42)^4 * (1 - 14/42)^(5 - 4)
= 5 * (14/42)^4 * (2/3)^1
= 80/243

For x = 5:
P(5 red candies) = (5C5) * (14/42)^5 * (1 - 14/42)^(5 - 5)
= 1 * (14/42)^5 * (2/3)^0
= (1/3)^5
= 1/243

Now, let's calculate the expected number of red candies:

Expected number of red candies = (0 * P(0 red candies)) + (1 * P(1 red candy)) + (2 * P(2 red candies)) + (3 * P(3 red candies)) + (4 * P(4 red candies)) + (5 * P(5 red candies))
= (0 * 32/243) + (1 * 70/81) + (2 * 280/243) + (3 * 320/243) + (4 * 80/243) + (5 * 1/243)
= 70/243 + 560/243 + 960/243 + 320/243 + 80/243 + 5/243
= 1995/243
≈ 8.21

Therefore, the expected number of red candies among 5 candies poured from the box is approximately 8.21.