consider a scenario where a binomial experiment is conducted and the probability of success in each trial is 40%. if the experiment includes 15 trials, what is the probability that out of those 15 trials, exactly 10 will be successful?
Pls help with formula!
C(15,10) (.4)^10 (.6)^4
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omg... the moment i visit this site and the moment i revisit i see a 1 ...
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To find the probability of exactly 10 successes in 15 trials of a binomial experiment, you can use the binomial probability formula.
The binomial probability formula is as follows:
P(x) = (nCx) * p^x * q^(n-x)
Where:
- P(x) represents the probability of getting exactly x successes,
- n is the total number of trials,
- x is the number of successes you want,
- p is the probability of success in each trial, and
- q is the probability of failure in each trial (1 - p).
In this scenario:
- n = 15 (total number of trials),
- x = 10 (number of successful trials),
- p = 0.40 (probability of success in each trial), and
- q = 1 - p = 1 - 0.40 = 0.60 (probability of failure in each trial).
Now, plug these values into the formula:
P(10) = (15C10) * (0.40)^10 * (0.60)^(15-10)
To calculate (15C10), which represents the number of ways to choose 10 successes out of 15 trials, you can use the binomial coefficient formula:
(15C10) = 15! / (10! * (15-10)!)
Calculating the binomial coefficient:
(15C10) = 15! / (10! * 5!)
Now, calculate each part of the formula:
(15C10) = 15! / (10! * 5!) = (15 * 14 * 13 * 12 * 11) / (5 * 4 * 3 * 2 * 1) = 3003
Next:
P(10) = 3003 * (0.40)^10 * (0.60)^(15-10) = 0.23622 (rounded to five decimal places)
Therefore, the probability that exactly 10 out of 15 trials will be successful is approximately 0.23622, or 23.62%.