Question 1029767: Could you please help me with this question?
The door prizes at a dance are four $10 gift certificates, five $20 gift certificates, and three $50 gift certificates. The prize envelopes are mixed together in a bag, and five prizes are drawn at random.
What is the probability that none of the prizes is a $10 gift certificate?
What is the expected number of $20 gift certificates drawn?
Answer by mathmate(429)   (Show Source):
You can put this solution on YOUR website!
Question:The door prizes at a dance are four $10 gift certificates, five $20 gift certificates, and three $50 gift certificates. The prize envelopes are mixed together in a bag, and five prizes are drawn at random.
(a) What is the probability that none of the prizes is a $10 gift certificate?
(b) What is the expected number of $20 gift certificates drawn?
Solution:
This is a typical problem of picking objects from a mixture of "good" and "bad" objects without replacement. The modelling distribution is the hypergeometric distribution, given by:
P(a)=C(A,a)*C(B,b)/C(A+B,a+b)
where
a=number of "good" objects picked
b=number of "bad" objects picked.
A=number of "good" objects in the given batch, and
B=number of "bad" objects in the given batch.
a+b=total number of objects picked (without replacement)
A+B=total number of objects in the batch.
(a) P($10=0
Here A=4 ($10, good), B=3+5=8 ($20, $50, "bad")
a=0, b=5 (remember, a+b=5)
therefore
P(a=0)=C(4,0)*C(8,5)/C(12,5)
=1*56/792
=7/99
(b) Find E[$20]
$20 is the "good" prize.
A=5, B=3+4=7, a+b=5 (total number of prizes drawn)
E[$20]=(5/(5+3+4)*5)=(5/12)*5=25/12=2.083 (to 3 decimal places)
Expected value is equation to (A/(A+B))*(a+b)
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