SOLUTION: Probability in three independent events, that event occurs atleast once is 0.657. Find probability that in 5 experiments the event will occur at least 4 times.
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-> SOLUTION: Probability in three independent events, that event occurs atleast once is 0.657. Find probability that in 5 experiments the event will occur at least 4 times.
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Question 1198527: Probability in three independent events, that event occurs atleast once is 0.657. Find probability that in 5 experiments the event will occur at least 4 times. Answer by CPhill(1959) (Show Source):
You can put this solution on YOUR website! **1. Determine the Probability of the Event Occurring in a Single Experiment**
* Let 'p' be the probability of the event occurring in a single experiment.
* We know that the probability of the event occurring at least once in three independent experiments is 0.657.
* This can be expressed as: 1 - (1 - p)³ = 0.657
* Solving for 'p':
* (1 - p)³ = 1 - 0.657 = 0.343
* 1 - p = ∛0.343
* 1 - p = 0.7
* p = 0.3
**2. Use the Binomial Probability Formula**
* We'll use the binomial probability formula to calculate the probability of the event occurring at least 4 times in 5 experiments.
* The binomial probability formula is:
* P(X = k) = (nCk) * p^k * (1-p)^(n-k)
* where:
* n is the number of trials (5 experiments)
* k is the number of successes (4 or 5 events occurring)
* p is the probability of success in a single trial (0.3)
* (nCk) is the number of combinations of n things taken k at a time
* **Probability of the event occurring exactly 4 times:**
* P(X = 4) = (5C4) * (0.3)^4 * (0.7)^(5-4) = 5 * 0.0081 * 0.7 = 0.02835
* **Probability of the event occurring exactly 5 times:**
* P(X = 5) = (5C5) * (0.3)^5 * (0.7)^(5-5) = 1 * 0.00243 * 1 = 0.00243
* **Probability of the event occurring at least 4 times:**
* P(X ≥ 4) = P(X = 4) + P(X = 5) = 0.02835 + 0.00243 = 0.03078
**Therefore, the probability that the event will occur at least 4 times in 5 experiments is approximately 0.0308.**
