Question

7. If you flip 10 fair coins, what is the probability that: (a) Exactly 5 of them are “heads”? (b) At

least 8 of them are “heads”?

8. Urn A has four red balls and two white balls, and urn B has three red balls and four white

balls. A fair coin is tossed. If it lands heads up, a ball is drawn from urn A; otherwise, a ball is

drawn from urn B. Compute (a) the probability a red ball is drawn (b) the probability that the

coin landed heads up, if it is known that a red ball was drawn.

9. If two events are disjoint, can they be independent? If yes, give an example; if no, prove that it

is not possible.

13. Suppose that ? is a discrete random variable with ?(? = 0) = 0.3, ?(? = 1) = 0.2, ?(? =

2) = 0.15, and ?(? = 3) = 0.35. Graph the frequency function and the cumulative distribution

function of ?.

15. 80 percent students drive to campus. Which is more likely (a) Out of 10 randomly

selected students, exactly 5 drive to campus (b) Out of 20 randomly selected

students, exactly 10 drive to campus.

16. If ? is a geometric random variable with ? = 0.5, for what value of ? is ?(? ≤ ?) ≃ 0.95?

Answer 1

7)

Sample size , n =    10
Probability of an event of interest,P(head) = p =   0.5

a) P ( X =    5   ) = C(10,5) * 0.5^5 * (1-0.5)^5 =            0.2461   (answer)
b)

P ( X = 8) = C (10,8) * 0.5^8 * ( 1 - 0.5)^2=      0.0439
P ( X = 9) = C (10,9) * 0.5^9 * ( 1 - 0.5)^1=      0.0098
P ( X = 10) = C (10,10) * 0.5^10 * ( 1 - 0.5)^0=      0.0010

P(x≥8) = P(x=8) + P(X=9) + P(x=10 ) = 0.0546875

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8)

A- 4 R, 2 W  
B - 3R , 4W  
  
a) P(red ball) = P(urn A and red ball) + P(urn b and red ball) = 0.5*4/6+0.5*3/7=   0.5476
(b) the probability that the   coin landed heads up, if it is known that a red ball was drawn =P(urn A | red ball) =(0.5*4/6)/0.5476=   0.6087