A pharmaceutical company knows that approximately 5% of its birth-control pills have an ingredient that is below the minimum strength, thus rendering the pill ineffective. What is the probability that fewer than 10 in a sample of 200 pills will be ineffective
Answers
Answer:
The probability that fewer than 10 in a sample of 200 pills will be ineffective is 0.4364
∴ P(X<10)=0.4364
Step-by-step explanation:
Given that the total number of sample pills is 200
ie., n=200
To find the probability that fewer than 10 in a sample of 200 pills will be ineffective:
Let us assume it success if a pill is ineffective.
The probability of success in each trial is
∴ p=0.05
We know that the total probability is p+q=1
The probability of failure is q
q=1-p
q=1-0.05
∴ q=0.95
Let X be the random variable of the number of ineffective pills in a sample of 200 pills.
Hence X has Binomial distribution with parameter n=200 and p=0.05
The formula for Mean in Binomial distribution is
Substitute the values in the above formula we get
∴
The formula for Standard deviation in Binomial distribution is
Substitute the values in the above formula we get
∴
Now we have to find the probability that fewer than 10 in a sample of 200 pills will be ineffective.
That is to find the area to the left of x=9.5
The formula is
Substitute the values in the formula we get
∴
Now P(X<10)=P(Z<-0.16)
=0.4364
∴ P(X<10)=0.4364
Therefore the probability that fewer than 10 in a sample of 200 pills will be ineffective is 0.4364
Final answer:
The probability that fewer than 10 out of 200 birth-control pills will be ineffective is approximately 0.817, or 81.7%.
Explanation:
Probability is a mathematical concept used to quantify the likelihood of an event occurring. It ranges from 0 (impossible) to 1 (certain). It plays a crucial role in statistics and decision-making, helping to predict outcomes, assess risk, and make informed choices. To find the probability that fewer than 10 in a sample of 200 birth-control pills will be ineffective, we can use the binomial probability formula:
P(X < 10) = Σ (n choose k) * p^k * (1-p)^(n-k), where:
n = sample size = 200,
k = number of ineffective pills,
p = probability of a pill being ineffective = 0.05.
Calculating this probability using the formula, we get:
P(X < 10) ≈ 0.817, or 81.7%.
Learn more about Probability here:
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