Test the potency of a drug on 20 rats. Previous animal show that a 10-mg dose of drug is lethal 5% of the time within 1st 4 hours; of the animals alive at 4 hours, 10% will die in the next 4 hours. What is the probability that 3 or more rats will die in the first 4 hours.
To determine the probability that 3 or more rats will die within the first 4 hours, we need to calculate the cumulative probability of the outcome using the given data.
Firstly, we need to determine the probability of a rat surviving the initial 4 hours. We know that a 10-mg dose of the drug is lethal 5% of the time within the first 4 hours. Therefore, the probability of a rat surviving the initial 4 hours is 1 - 0.05 = 0.95.
Now, we need to calculate the probability of a rat dying in the first 4 hours. Since we have 20 rats, we'll use the binomial probability formula:
P(x) = nCx * p^x * q^(n-x)
Where:
- P(x) is the probability of x rats dying in the first 4 hours
- n is the total number of rats (20)
- x is the number of rats dying in the first 4 hours (3 or more)
- p is the probability of a rat dying in the first 4 hours (0.05)
- q is the probability of a rat surviving the first 4 hours (0.95)
We need to calculate P(x) for x = 3, 4, 5, ..., 20, and sum up these probabilities to get the final answer.
P(3 or more rats dying) = P(3) + P(4) + P(5) + ... + P(20)
Using a statistical software or calculator with a binomial probability function, you can easily calculate the probabilities for each value of x and sum them up to find the final answer.
Alternatively, you can use Excel or Google Sheets to calculate the probabilities using the BINOM.DIST function:
=BINOM.DIST(3, 20, 0.05, TRUE) + BINOM.DIST(4, 20, 0.05, TRUE) + ... + BINOM.DIST(20, 20, 0.05, TRUE)
This will give you the probability that 3 or more rats will die in the first 4 hours.