The probability of a child getting coal in their stocking is 0.2. If Santa visits 3 kids on Christmas day a) what is the probability of 2 children getting coal? b) 1 or fewer children receiving coal? Please help me - my brain is fried!

Sure! I'd be happy to help you with these probability questions. To find the probability of events, we can use the concept of independent events. Each child getting coal or not getting coal is considered an independent event because the outcome for one child does not affect the outcome for the other children.

a) To find the probability of 2 children getting coal, we can use the binomial probability formula, which is:

P(X=k) = (nCk) * p^k * (1-p)^(n-k)

In this case, n represents the number of trials (which is 3, since Santa visits 3 kids), k represents the number of successes (which is 2, since 2 children getting coal), p represents the probability of success (which is 0.2, since there is a 0.2 probability of a child getting coal), and (nCk) represents the combination formula n! / (k!(n-k)!).

Applying these values to the formula, we have:

P(X=2) = (3C2) * (0.2)^2 * (1-0.2)^(3-2)

Calculating this, we get:

P(X=2) = (3! / (2!(3-2)!)) * (0.2)^2 * (0.8)^1

P(X=2) = 3 * 0.04 * 0.8

P(X=2) = 0.096

Therefore, the probability of 2 children getting coal is 0.096 or 9.6%.

b) To find the probability of 1 or fewer children receiving coal, we need to find the probability of 0 children getting coal and the probability of 1 child getting coal, and then add those probabilities together.

The probability of 0 children getting coal can be calculated using the formula:

P(X=0) = (3C0) * (0.2)^0 * (1-0.2)^(3-0)

Calculating this, we get:

P(X=0) = (3! / (0!(3-0)!)) * (0.2)^0 * (0.8)^3

P(X=0) = 1 * 1 * 0.512

P(X=0) = 0.512

The probability of 1 child getting coal can be calculated using the formula:

P(X=1) = (3C1) * (0.2)^1 * (1-0.2)^(3-1)

Calculating this, we get:

P(X=1) = (3! / (1!(3-1)!)) * (0.2)^1 * (0.8)^2

P(X=1) = 3 * 0.2 * 0.64

P(X=1) = 0.384

Therefore, the probability of 1 or fewer children receiving coal is:

P(X<=1) = P(X=0) + P(X=1) = 0.512 + 0.384 = 0.896 or 89.6%

I hope this explanation helps! Let me know if you have any further questions.