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 probabilities, we can use the binomial probability formula. The binomial probability formula calculates the probability of a certain number of successes in a fixed number of independent trials, where each trial has the same probability of success.
In this case, the probability of a heart disease death is 1 out of 4 deaths, or 1/4. The total number of trials is 25.
To calculate the probability that exactly 5 of the deaths will be due to heart disease, we can use the binomial probability formula:
P(X = k) = (n C k) * p^k * q^(n-k)
Where:
- P(X = k) is the probability of exactly k successes
- n is the total number of trials
- k is the number of successes
- p is the probability of success
- q is the probability of failure, which is 1-p
Using these values, we can calculate:
P(X = 5) = (25 C 5) * (1/4)^5 * (3/4)^(25-5)
To calculate the probability that at most 4 of the deaths will be due to heart disease, we need to sum up the individual probabilities for 0, 1, 2, 3, and 4 deaths due to heart disease. We can use the same binomial probability formula for each case and add them together.
P(X ≤ 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)
Calculating each individual term using the formula and adding them together will give us the probability.
Note that (n C k) represents the combination formula, also known as the binomial coefficient, which calculates the number of ways to choose k objects out of a set of n objects.
Let's calculate the probabilities.