Lead Contamination More than a decade ago high levels of lead in the blood put 88% of children at risk. A concerted effort was made to remove lead from the environment.

In a random sample of 200 children taken more than a decade ago, what is the probability that 50 or more had high blood-lead levels?

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To find the probability that 50 or more children had high blood-lead levels in a random sample of 200 children taken more than a decade ago, we can use the binomial distribution formula.

The binomial distribution formula is given by:

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

Where:
P(X = k) represents the probability of getting exactly k successes (in this case, k children with high blood-lead levels)
n is the number of trials (sample size)
p is the probability of success in each trial

In this case, n = 200 (the sample size) and we need to find the probability of having 50 or more children with high blood-lead levels, so we need to calculate the sum of probabilities from 50 to 200.

Let's break down the steps to find the probability:

Step 1: Determine the probability of success (p)
The problem states that more than a decade ago, high levels of lead in the blood put 88% of children at risk. So the probability of a child having high blood-lead levels is 0.88.

Step 2: Calculate the probability using the binomial distribution formula
P(X >= 50) = P(X = 50) + P(X = 51) + ... + P(X = 200)

We can calculate each individual probability using the formula mentioned earlier.

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

For example, to calculate P(X = 50), plug in the values:
k = 50
n = 200
p = 0.88
Calculate (nCk) using the formula (n! / (k! * (n-k)!)):

(NOTE: Calculating probabilities for 51 to 200 can be time-consuming, but the steps are the same.)

Step 3: Sum up the probabilities from k = 50 to k = 200
Calculate the sum of the probabilities you obtained in step 2. This will give you the probability that 50 or more children had high blood-lead levels in the random sample of 200 children taken more than a decade ago.

P(X >= 50) = P(X = 50) + P(X = 51) + ... + P(X = 200)

Once you have calculated these individual probabilities, simply add them together to get the final probability.

Note: You may want to use statistical software or a calculator with binomial distribution functionality to help with the calculations, as they can be time-consuming if done manually.