Question 425117: Two cards are drawn, with replacement, from a standard 52-card deck. Find the expected number of diamonds
Found 2 solutions by stanbon, Edwin McCravy:
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! Two cards are drawn, with replacement, from a standard 52-card deck. Find the expected number of diamonds
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On each draw the expected # of diamonds is (1/4)(13) = 3.25
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Cheers,
Stan H.
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Answer by Edwin McCravy(20064)  (Show Source):
You can put this solution on YOUR website!
The other tutor's answer is incorrect.
The possibilities are
1. 0 diamonds, that is, a non-diamond the first time and
a non-diamond the second time.
2. 1 diamond, that is, a diamond the first time and a non-diamond
the second time, or a non-diamond the first time and a diamond
the second time.
3. 2 diamonds, that is, a diamond the first time and a diamond
the second time.
Remember that "AND" usually indicates that you are to multiply
probabilities, while "OR" indicates that you are to add them.
The probability of drawing a diamond is 1/4 and the probability of
drawing a non-diamond is 3/4
1. Find the probability of 0 diamonds, (two non-diamonds)
P(non-diamond first AND non-diamond second) =
P(non-diamond first)*P(non-diamond second)
(3/4)(3/4) = 9/16
2. Find the probability of exactly 1 diamond (and 1 non-diamond)
P[(diamond first AND non-diamond second) OR (non-diamond first
AND diamond second) =
P(diamond first)*P(non-diamond second) +
P(non-diamond first)*P(diamond second) =
(1/4)(3/4) + (3/4)(1/4) = 3/16 + 3/16 = 6/16 = 3/8
3. Find the probability of 2 diamonds:
P(diamond first AND diamond second) =
P(diamond first)*P(diamond second) =
(1/4)(1/4) = 1/16
Next we make a discrete probability distribution function:
Number of
diamonds Probability
x P(x)
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0 9/16
1 3/8
2 1/16
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1
Notice that the sum of the probabilities of all possible events is
9/16 + 3/8 + 1/16 = 9/16 + 6/16 + 1/16 = 16/16 = 1,
which you should always check.
To find the expectation of the variable x:
E(x) = ∑[x*P(x)] = 0*(9/16) + 1*(3/8) + 2*(1/16) =
0 + 3/8 + 2/16 = 3/8 + 1/8 = 4/8 = 1/2
So the expectation is 1/2.
[Note: "Expectation" means what you would expect to average if
you repeated the experiment every day for a long period of time
and took the average of the number of diamonds you drew. It does
NOT mean that you would ever "expect" to get half a diamond.
However if you drew two cards without replacement every day for
365 days, you might get say 181 diamonds, and you would have
averaged 181/365 or .49589 diamonds per day, which is very near .5
or 1/2.]
Edwin
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