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kfhayworth4480
04.04.2020 •
Mathematics
Suppose we roll a fair six-sided die 20 times and draw ten cards from a standard 52-card deck. Let X be the number of "6"s rolled plus the number of Jack, Queen, King, or Aces drawn (There are 16 such cards in the 52).
(a) Calculate the Expected value, Variance, and Standard deviation of X.
Hint: Let X1 be the number of "6"s rolled and X2 be the number of Jacks or better drawn. Then, X = X1 +X2, and X1 and X2 are independent.
(b) What is the probability that we roll at least five "6"'s and, at the same time, draw at least 4 Jacks, Queens, Kings, or Aces?
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Ответ:
a) Expected value = 6.406
Variance = 4.905
Standard deviation = 2.45
b) The probability is 0.08547
Step-by-step explanation:
a) Let's suppose that:
X₁ = number of 6´s
X₂ = number of Jack, Queen, King or Aces
The mean of X₁ is:
MeanX₁ = n * p = 20 * (1/6) = 3.33
The variance of X₁ is:
The mean of X₂ is:
MeanX₂ = 10 * (16/52) = 3.076
The variance of X₂ is:
The expect value of X is:
Xexp = MeanX₁ + MeanX₂ = 3.33 + 3.076 = 6.406
The variance of X is:
VarX = VarX₁ + VarX₂ = 2.775 + 2.13 = 4.905
The standard deviation is:
Xdevi = 4.905/2 = 2.45
b) The probability of drawing at least five six out of 20 rolls is equal to:
∑(1/6)ˣ(5/6)²⁰⁻ˣ = 0.231 with x = 5
The probability of at least 4 Jack, Queen, Kings or Aces is:
∑(16/52)ˣ(1-(16/52))¹⁰⁻ˣ = 0.37 with x = 4
The probability of given event is equal to:
P = 0.231 * 0.37 = 0.08547
Ответ:
Then the amount invested at 9% is 6000 - x.
0.05x + 0.09(6000 - x) = 380
Solving the equation gives: x = 4000.
The answer is: $4000 invested at 5% and $2000 invested at 9%.