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Answer:
The probability is 0.0775
The expected value is 4.75 clubs
The standard deviation is 1.8875 clubs
Step-by-step explanation:
The variable X follows a binomial distribution, because we have n identical and independent events (19 cards) with a probability p of success and 1-p of fail (there is a probability of 1/4 to be club and 3/4 to be diamond, heart or spade). Then, the probability that x of the n cards are club is:
So, the probability P of drawing at least 8 clubs is:
P = P(8) + P(9) + P(10) + ... + P(18) + P(19)
Replacing, the values of x, from 8 to 19, on the equation above, we get:
P = 0.0775
Additionally, the expected value E(x) and standard deviationS(x) for this distribution is given by:
E(x)=np = 19(0.25) = 4.75