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Practice
Problems in Probability
Easy
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Problems in Probability: Difficult Problems with Solutions
Problem 1
Let S={a, b, c} find the value of P(b') if P(b')=1-P(b), P(a)=2P(b) and P(c)=¼
Answer format: x/y
Solution:
P(a)+P(b)+P(c)=1 => 2P(b)+P(b)+1/4=1
P(b)=1/4
P(b')=1-P(b)=1-1/4=3/4
Problem 2
Suppose we have 5 red balls and 4 black balls and get 5 balls randomly. What's the probability of getting 3 red balls and 2 black balls?
Answer format: x/y
Solution:
Let's denote the event of getting 3 red balls and 2 black balls by A, so
[tex]P(A)=\frac{n(A)}{n(S)}=\frac{ {{5}\choose{3}} \times {{4}\choose{2}} } { {9}\choose{5} }=\frac{10\times6}{126}=\frac{30}{63}[/tex]
Problem 3
Choose two real number between 0 and 2 at random. Find the probability that the summation of these two numbers is less than 2.
Answer format: x/y
Solution:
S={(x,y)|0< x < 2, 0 < y <2}
A={(x,y)|0< x,y < 2, x+y <2}
[tex]P(A)=\frac{a_A}{a_S}=\frac{\frac{1}{2}\times2\times2}{2\times2}=\frac{1}{2}[/tex]
Problem 4
Suppose we have a dice that the probability of odd numbers is 5 times more than even numbers.
Find the probability that dice shows 6.
Answer format: x/y
Solution:
P(1)=P(3)=P(5)=5P(2)=5P(4)=5P(6)
P(1)+P(2)+P(3)+P(4)+P(5)+P(6)=1
5P(6)+P(6)+5P(6)+P(6)+5P(6)+P(6)=1
18P(6)=1
P(6)=1/18
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