I have gotten the concept of probability but the following question has really stumped me. Help please.
The probability that a tomato seed will germinate is 0.9. Estimate the probability that of 100 randomly selected seeds, at least 86, but no more than 91 of them will germinate.
100 trials
probability of success = .9
probability of failure = .1
we need P(86)
P(87)
P(88)
etc down to
P(91)
This is a binary distribution problem
Then add them all
I will do one of them
P(86) = 100C86 * .9^86 * .1^(100-86)
100C86 = 100! /[ 86!(14!)]
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HOWEVER
as you get a large number of trials the binomial distribution converges to a normal distribution with
mean = n p
in this case mean = 100*.9 = 90
nd sigma = sqrt (n p (1-p) )
sqrt (9) = 3
well that was easy
now go to a normal distribution table
look for 86 to 91 with a mean of 90 and a sigma of 3
try
http://davidmlane.com/hyperstat/z_table.html
I get .5393
To estimate the probability in this scenario, we can use the binomial probability formula. The formula uses the probability of success (germination) for one trial, the number of trials, and the desired range of successful outcomes.
The binomial probability formula is:
P(x) = (nCx) * p^x * (1-p)^(n-x)
Where:
P(x) is the probability of x successful outcomes,
n is the number of trials,
x is the desired number of successful outcomes,
p is the probability of success for one trial, and
(nCx) represents the number of combinations (also known as the binomial coefficient) of n items taken x at a time.
In this case, we want to calculate the probability of at least 86 but no more than 91 seeds germinating out of 100 trials, with a success probability of 0.9 (given in the question).
To calculate this range, we need to find the sum of probabilities for each individual outcome from 86 to 91. We can write it as:
P(86 or 87 or 88 or 89 or 90 or 91) = P(86) + P(87) + P(88) + P(89) + P(90) + P(91)
Now, let's calculate each individual probability step-by-step:
P(86) = (100C86) * 0.9^86 * (1-0.9)^(100-86)
P(87) = (100C87) * 0.9^87 * (1-0.9)^(100-87)
P(88) = (100C88) * 0.9^88 * (1-0.9)^(100-88)
P(89) = (100C89) * 0.9^89 * (1-0.9)^(100-89)
P(90) = (100C90) * 0.9^90 * (1-0.9)^(100-90)
P(91) = (100C91) * 0.9^91 * (1-0.9)^(100-91)
Finally, we can add up these individual probabilities to get the estimate of the desired probability:
P(at least 86 but no more than 91) = P(86) + P(87) + P(88) + P(89) + P(90) + P(91)
Please note that calculating this manually might be time-consuming. You could use a calculator, spreadsheet, or statistical software that supports binomial probability calculations to simplify the process.