You are responsible for coordinating an air drop of supplies to some sailors who are stranded on a remote, Pacific island. You plan to deliver 10 packages, each with its own parachute. The probability that any single package will make it to the island (and not get swept off course due to winds) is 30%. The stranded sailors need at least 4 packages in order to survive.What is the probability that the sailors will survive? Round your answer to the nearest hundredth place.
Question
You are responsible for coordinating an air drop of supplies to some sailors who are stranded on a remote, Pacific island. You plan to deliver 10 packages, each with its own parachute. The probability that any single package will make it to the island (and not get swept off course due to winds) is 30%. The stranded sailors need at least 4 packages in order to survive.What is the probability that the sailors will survive? Round your answer to the nearest hundredth place.
Solution
This problem can be solved using the binomial probability formula, which is:
P(X=k) = C(n, k) * (p^k) * ((1-p)^(n-k))
where:
- P(X=k) is the probability of k successes in n trials
- C(n, k) is the number of combinations of n items taken k at a time
- p is the probability of success on a single trial
- n is the number of trials
- k is the number of successes
In this case, we want to find the probability that at least 4 packages make it to the island. This is the same as finding the probability that 4, 5, 6, 7, 8, 9, or 10 packages make it to the island. So, we need to calculate each of these probabilities and then add them together.
Let's start with the probability that exactly 4 packages make it to the island:
P(X=4) = C(10, 4) * (0.3^4) * ((1-0.3)^(10-4))
Next, we calculate the probabilities for 5, 6, 7, 8, 9, and 10 packages and add all these probabilities together to get the total probability that at least 4 packages make it to the island.
The final answer will be the sum of these probabilities, rounded to the nearest hundredth place.
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