According to the American Heart Association, one in every four deaths in the U.S. is due to heart disease. Suppose 25 people will die in the U.S. in the next 5 minutes.
• What is the chance that exactly 5 of the deaths will be due to heart disease?
• What is the chance that at most 4 of the deaths will be due to heart disease?
To find the probability of these scenarios, we need to use the binomial probability formula. This formula is used to calculate the probability of a specific number of successes in a fixed number of independent Bernoulli trials, where the probability of success remains constant.
The formula for the probability of getting exactly k successes in n trials, with a probability of success p, is:
P(X = k) = (n choose k) * p^k * (1-p)^(n-k)
where (n choose k) represents the number of ways to choose k successes from n trials and is given by the binomial coefficient:
(n choose k) = n! / (k! * (n-k)!)
Let's use this formula to calculate the probabilities:
1. Probability of exactly 5 deaths due to heart disease:
In this case, we have n = 25 trials (deaths in the next 5 minutes) and p = 1/4 (probability of a death being due to heart disease). To calculate the probability:
P(X = 5) = (25 choose 5) * (1/4)^5 * (3/4)^(25-5)
Using a calculator or a computer program, we can evaluate this expression to find the probability.
2. Probability of at most 4 deaths due to heart disease:
To calculate this probability, we need to sum up the probabilities of having 0, 1, 2, 3, or 4 deaths due to heart disease. This can be done using the binomial probability formula for each value and then summing them up:
P(X <= 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)
To calculate each individual probability, we use the same formula as above, plugging in the corresponding values for n and p.
Again, using a calculator or a computer program, we can evaluate each individual probability and then add them up to find the desired probability.