Probability of getting 8 or more diamonds = 0.00278151
Mean and standard deviation = 3 and 1.5
Since you're replacing the card and shuffling, each event is independent and has a probability of 1/4 of success and 3/4 of failure.
You succeed if you pick 8, 9, 10, 11, or 12 diamonds.
For picking 8 diamonds, there are
12!(8!4!) ways of picking exactly 8 diamonds and the probability of each of those possibilities is 0.25^8 * 0.75^4, so the total probability of exactly 8 diamonds is
12!(8!4!)*0.25^8*0.75^4 = 0.002389848
Continuing with the same logic, the probabilites for 9, 10, 11, and 12 diamonds is
12!(9!3!)*0.25^9*0.75^3 = 0.000354052
12!(10!2!)*0.25^10*0.75^2 = 0.0000354052
12!(11!1!)*0.25^11*0.75^1 = 0.00000214577
12!(12!0!)*0.25^12*0.75^0 = 0.0000000596046
And the total probability is the sum of those 5 terms. Giving
0.002389848 + 0.000354052 + 0.0000354052 + 0.00000214577 + 0.0000000596046 = 0.00278151
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This is a problem with a binomial distribution, with p = 0.25 and n = 12.
The expected value of X is simply n times p, so
12 * 0.25 = 3
The standard deviation is expressed as
d = sqrt(np(1-p))
where
d = standard deviation
n = number of selections
p = probability of selection
Substituting known values, we get
d = sqrt(12 * 0.25(1-0.25))
d = sqrt(12 * 0.25 * 0.75)
d = sqrt(2.25)
d = 1.5
Word of warning, the formula doesn't closely approximate a standard deviation curve unless the following is true.
n > 9(1-p)/p and n > 9p/(1-p)
With the p value of 0.25, the limits are
9(1-0.25)/0.25 = 9*0.75/0.25 = 27
and
9*0.25/(1-0.25) = 9*0.25/0.75 = 3
So n should be at least 28 and 12 is rather small for a proper distribution.
