The survival rate during a risky operation for patients with no other hope of survival is 81%. What is the probability that exactly four of the next five patients survive this operation?
To calculate the probability that exactly four out of the next five patients survive this risky operation, we need to use the concept of binomial probability. The binomial probability measures the likelihood of a certain number of successes in a fixed number of independent experiments.
In this case, the probability of survival for each patient is 81%, which means the probability of not surviving is 19%. Since the probability of success (survival) is constant for each patient, the situation can be modeled as a binomial distribution.
The formula to calculate the probability of exactly "k" successes in "n" independent trials is:
P(X = k) = combination(n, k) * p^k * (1-p)^(n-k)
Where:
- "P(X = k)" represents the probability of getting exactly "k" successes
- "combination(n, k)" represents the number of ways to choose "k" successes from "n" trials
- "p" represents the probability of success (survival) for each trial
- "1-p" represents the probability of failure (not surviving) for each trial
- "^" denotes exponentiation
In this case, since we want to calculate the probability of exactly four out of the next five patients surviving, we substitute:
n = 5
k = 4
p = 0.81 (probability of survival)
1-p = 0.19 (probability of not surviving)
Now, let's plug in the values and calculate:
P(X = 4) = combination(5, 4) * 0.81^4 * 0.19^(5-4)
Combination(5, 4) is equal to 5, as there are five ways to choose four successes out of five trials.
P(X = 4) = 5 * 0.81^4 * 0.19^1
P(X = 4) = 5 * 0.531441 * 0.19
P(X = 4) ≈ 0.5067
Therefore, the probability that exactly four out of the next five patients survive this operation is approximately 0.5067, or 50.67%.
To calculate the probability that exactly four of the next five patients survive this operation, we can use the binomial probability formula.
The formula for binomial probability is:
P(x) = C(n, x) * p^x * (1-p)^(n-x)
Where:
P(x) is the probability of exactly x successes
C(n, x) is the number of combinations of n items taken x at a time
p is the probability of success
(1-p) is the probability of failure
n is the number of trials
In this case:
n = 5 (number of trials)
x = 4 (number of successes)
p = 0.81 (probability of success)
(1-p) = 0.19 (probability of failure)
Using the formula, we can calculate the probability as follows:
P(4) = C(5,4) * 0.81^4 * 0.19^(5-4)
Using the combination formula C(n, x) = n! / (x!(n-x)!), we get:
P(4) = (5! / (4!(5-4)!)) * 0.81^4 * 0.19^(5-4)
Simplifying:
P(4) = (5 * 4! / (4!(1!))) * 0.81^4 * (0.19^1)
P(4) = 5 * 0.81^4 * 0.19
P(4) = 5 * 0.531441 * 0.19
P(4) = 0.50626
Therefore, the probability that exactly four of the next five patients survive this operation is approximately 0.50626 or 50.6%.