(b) What is the probability that a randomly selected page that contains 30 lines

Question

(b) What is the probability that a randomly selected page that contains 30 lines has exactly four typos? Two fair dice are thrown. (a) Find the probability of getting a sum of 5 or more. (b) Find the probability of getting a sum of 5 or more given than the sum is even. (c) Are the two events "sum of 5 or more" and "sum is even independent? For f(x) = {cx^2(1 - x)^2, 0 lessthanorequalto x lessthanorequalto 1 0, elsewhere (a) Find c so that f(x) is a probability density function for some random variable x.

Explanation / Answer

Two fair dice are thrown

Probability of any number = 1/6

no of possibities for two die rolls are 36

possibilities of sum of two fair die rolls are 2,3,4,5,6,7,8,9,10,11,12

P(sum=2) = 1/36 i.e (1,1)

P(sum=3) = 2/36 i.e (1,2)(2,1)

P(sum=4) = 3/36 i.e (1,3)(2,2)(3,1)

P(sum=5) = 4/36 i.e (1,4)(2,3)(3,2)(4,1)

P(sum=6) = 5/36 i.e (1,5)(2,4)(3,3)(4,2)(5,1)

P(sum=7) = 6/36 i.e (1,6)(2,5)(3,4)(4,3)(5,2)(6,1)

P(sum=8) = 5/36 i.e (2,6)(3,5)(4,4)(5,3)(6,2)

P(sum=9) = 4/36 i.e (3,6)(4,5)(5,4)(6,3)

P(sum=10) = 3/36 i.e (4,6)(5,5)(6,4)

P(sum=11) = 2/36 i.e (5,6)(6,5)

P(sum=12) = 1/36 i.e (6,6)

a)

probability of getting a sum of 5 or more = P(5)+P(6)+P(7)+P(8)+P(9)+P(10)+P(11)+P(12)

= 4/36+5/36+6/36+5/36+4/36+3/36+2/36+1/36 = 0.8333

b)

Probabity of sum is even = P(2)+P(4)+P(6)+P(8)+P(10)+P(12)

= 1/36+3/36+5/36+5/36+3/36+1/36 = 0.5

Probabity of "sum of 5 or more" and "sum is even" = P(6)+P(8)+P(10)+P(12)

= 5/36+5/36+3/36+1/36 = 0.3889

Probabity of "sum of 5 or more" given that "sum is even" = 0.3889/0.5 = 0.7778

c)

Probabity of "sum of 5 or more" given that "sum is even" is not equal to Probabity of "sum of 5 or more"

so "sum of 5 or more" and "sum is even" not independent

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