Exercise 1.14. Assume that P(A) = 0.4 and P(B) = 0.7. Making no further assumpti

Question

Exercise 1.14. Assume that P(A) = 0.4 and P(B) = 0.7. Making no further assumptions on A and B, show that P(AB) satisfies 0.1 s PAB) 0.4. Exercise 1.15. An urn contains 4 balls: 1 white, 1 green and 2 red. We draw 3 balls with replacement. Find the probability that we did not see all three colors. Use two different calculations, as specified by (a) and (b) below. (a) Define the event W - (white ball did not appear) and similarly for G and R. Use inclusion-exclusion. (b) Compute the probability by considering the complement of the event that we did not see all three colors. Section 1.5 Exercise 1.16. We flip a fair coin five times. For every heads you pay me $1 and for every tails I pay you $1. Let X denote my net winnings at the end of five flips. Find the possible values and the probability mass function of X. Exercise 1.17. An urn contains 4 red balls and 3 green balls. Two balls are drawn randomly.

Explanation / Answer

(1.16)

Here X is the random variable for the net winnings.

The winnings can not be negative, so the only possible values for the net winnings are 1$, 3$ and 5$.

P(X=1) = P(3 tails and 2 heads) = 5C3*0.53*0.52 = 0.3125

P(X=3) = P(4 tails and 1 head) = 5C4*0.54*0.51 = 0.15625

P(X=5) = P(5 tails) = 5C5*0.53*0.52 = 0.03125

The probability values for net losses are also similar.

P(X=-1) = P(2 tails and 3 heads) = 5C3*0.53*0.52 = 0.3125

P(X=-3) = P(1 tails and 4 head) = 5C4*0.54*0.51 = 0.15625

P(X=-5) = P(5 heads) = 5C5*0.53*0.52 = 0.03125

Hope this helps !

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