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An infinite sequence of independent trials is to bo performed. Each Trial results in success with probability p and a failure with probability 1 - p. What is the probability that at least one success occurs in the first n trials?
- a)p(1-p)n-1
- b)(1-p)n
- c)1-(1-p)n
- d)pn
Correct answer is option 'C'. Can you explain this answer?
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An infinite sequence of independent trials is to bo performed. Each Tr...
1 - probability of no success in n trials = 1- (1 - p)n
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An infinite sequence of independent trials is to bo performed. Each Tr...
Problem:
An infinite sequence of independent trials is to be performed. Each trial results in success with probability p and a failure with probability 1 - p. What is the probability that at least one success occurs in the first n trials?
Solution:
To find the probability that at least one success occurs in the first n trials, we can use the complement rule. The complement of the event "at least one success" is the event "no success", i.e., all trials result in failure.
Complement Rule:
The complement of an event A is denoted by A' and is defined as everything that is not in A. The probability of an event A' is equal to 1 minus the probability of event A.
Calculating the Complement:
In this problem, the event "no success" is equivalent to having all n trials result in failure. The probability of the first trial resulting in failure is 1 - p. Since each trial is independent, the probability of all n trials resulting in failure is (1 - p)^n.
So, the probability of at least one success occurring in the first n trials is equal to 1 minus the probability of all n trials resulting in failure.
Calculations:
The probability of at least one success occurring in the first n trials = 1 - (1 - p)^n
Explanation of the Correct Answer:
The correct answer is option 'C' which states that the probability of at least one success occurring in the first n trials is 1 - (1 - p)^n. This is the correct answer because it follows the logic of the complement rule. The complement of the event "at least one success" is the event "no success", and the probability of the complement event is 1 minus the probability of the event itself.
Therefore, the probability of at least one success occurring in the first n trials is 1 - (1 - p)^n.
An infinite sequence of independent trials is to be performed. Each trial results in success with probability p and a failure with probability 1 - p. What is the probability that at least one success occurs in the first n trials?
Solution:
To find the probability that at least one success occurs in the first n trials, we can use the complement rule. The complement of the event "at least one success" is the event "no success", i.e., all trials result in failure.
Complement Rule:
The complement of an event A is denoted by A' and is defined as everything that is not in A. The probability of an event A' is equal to 1 minus the probability of event A.
Calculating the Complement:
In this problem, the event "no success" is equivalent to having all n trials result in failure. The probability of the first trial resulting in failure is 1 - p. Since each trial is independent, the probability of all n trials resulting in failure is (1 - p)^n.
So, the probability of at least one success occurring in the first n trials is equal to 1 minus the probability of all n trials resulting in failure.
Calculations:
The probability of at least one success occurring in the first n trials = 1 - (1 - p)^n
Explanation of the Correct Answer:
The correct answer is option 'C' which states that the probability of at least one success occurring in the first n trials is 1 - (1 - p)^n. This is the correct answer because it follows the logic of the complement rule. The complement of the event "at least one success" is the event "no success", and the probability of the complement event is 1 minus the probability of the event itself.
Therefore, the probability of at least one success occurring in the first n trials is 1 - (1 - p)^n.
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Solutions for An infinite sequence of independent trials is to bo performed. Each Trial results in success with probability p and a failure with probability 1 - p.What is the probability that at least one success occurs in the first n trials?a)p(1-p)n-1b)(1-p)nc)1-(1-p)nd)pnCorrect answer is option 'C'. Can you explain this answer? in English & in Hindi are available as part of our courses for Mathematics.
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