The survival rate during a risky operation for patients with no other hope of survival is 74%. What is the probability that exactly four of the next five patients survive this operation? (Give your answer correct to three decimal places.)
If the events are independent, the probability of both/all events occurring is determined by multiplying the probabilities of the individual events.
.74^4 * (1-.74) = ?
To find the probability that exactly four of the next five patients survive the operation, you can use the binomial probability formula.
The binomial probability formula is given by:
P(X=k) = (n C k) * (p^k) * (1-p)^(n-k)
Where:
- P(X=k) is the probability of getting exactly k successes in n trials,
- n is the number of trials (in this case, the number of patients, which is 5),
- k is the number of successes (in this case, the number of patients surviving, which is 4),
- p is the probability of success in a single trial (in this case, the survival rate, which is 74% or 0.74), and
- (n C k) represents the number of combinations of n items taken k at a time, which can be calculated using the formula: (n C k) = n! / (k! * (n-k)!)
Applying the formula to the given values, we have:
P(X=4) = (5 C 4) * (0.74^4) * (1-0.74)^(5-4)
Calculating the values:
(5 C 4) = 5! / (4! * (5-4)!) = 5
(0.74^4) ≈ 0.3162
(1-0.74)^(5-4) = 0.26
Now, substitute these values into the formula:
P(X=4) = 5 * 0.3162 * 0.26
Calculating this expression:
P(X=4) ≈ 0.4095
Therefore, the probability that exactly four of the next five patients survive the operation is approximately 0.4095 or 40.95%.