a medication is 75% effective against a bacterial infection. find the probability that if 12 people take the medication, at least 1 person"s infection will not improve

Formula:

P(x) = (nCx)(p^x)[q^(n-x)]

Find P(0), then subtract that value from 1.

n = 12
p = .25
q = 1 - p

I'll let you take it from here.

Good

To find the probability that at least 1 person's infection will not improve, we can use the complementary probability approach.

First, let's find the probability that a single person's infection will improve when taking the medication. Since the medication is 75% effective against the bacterial infection, the probability that a single person's infection will improve is 0.75.

The probability that a single person's infection will not improve is the complement of this, which is 1 - 0.75 = 0.25.

Now, let's find the probability that all 12 people's infections will improve when taking the medication. Since we assume the outcomes for different people are independent, we can multiply the probabilities together. So, the probability that all 12 people's infections will improve is (0.75)^12 = 0.0563.

Finally, we can find the probability that at least 1 person's infection will not improve by subtracting the probability that all infections will improve from 1. So, the probability that at least 1 person's infection will not improve is 1 - 0.0563 ≈ 0.9437, or approximately 94.37%.

Therefore, the probability that if 12 people take the medication, at least 1 person's infection will not improve is approximately 94.37%.

To find the probability that at least one person's infection will not improve out of 12 people taking the medication, we can use the concept of the complementary probability.

First, let's find the probability that a single individual's infection improves when taking the medication. We are given that the medication is 75% effective, so the probability of improvement for a single person is 0.75.

Now, let's find the probability that a single individual's infection does not improve. This is simply the complement of the probability of improvement, which is 1 - 0.75 = 0.25.

Next, we calculate the probability that all 12 individuals' infections improve. Since each individual's outcome is independent, we can multiply the probabilities together. Therefore, the probability that all 12 individuals' infections improve is (0.75)^12.

Finally, to find the probability that at least one person's infection will not improve, we subtract the probability that all 12 individuals' infections improve from 1 (complementary probability). So the probability we seek is 1 - (0.75)^12.

Calculating this expression gives us the final answer:

1 - (0.75)^12 ≈ 0.9795

Therefore, the probability that at least one person's infection will not improve out of 12 people taking the medication is approximately 0.9795, or 97.95%.