Solution of QUIZ 2. M262 Section 1. (21.02.2022).

Solution of QUIZ 2. M262 Section 1. (21.02.2022).


1. A box contains four coins: two fair coins, a two-headed coin and a coin
which shows up heads with probability 1/11. We randomly select one of the coins
and toss it. Given that it shows up heads find the probability that it is not a fair coin.

Solution: By Bayes’ formula
1/4 · 1 + 1/4 · 1/11 12
P = = .
1/4 · 1/2 + 1/4 · 1/2 + 1/4 · 1 + 1/4 · 1/11 23




2. A box contains 2 white and 3 red balls. We draw balls from the box one-by-one
without replacement. Let X be the total number of trials until the total number of
obtained red balls exceeds the total number of obtained white balls. Find V arX.

Solution: R(X) = {1, 3, 5}. fX (1) = 3/5, fX (3) = 2/5 · 3/4 · 2/3 = 1/5 and

fX (5) = 2/5 · 1/4 · 3/3 · 2/2 · 1/1 + 2/5 · 3/4 · 1/3 · 2/2 · 1/1 = 1/5.

Therefore,

EX = 1 · 3/5 + 3 · 1/5 + 5 · 1/5 = 11/5,

EX 2 = 12 · 3/5 + 32 · 1/5 + 52 · 1/5 = 37/5

and

V arX = 37/5 − (11/5)2 = 64/25.




1




This study source was downloaded by 100000899606396 from CourseHero.com on 09-25-2025 13:16:03 GMT -05:00


https://www.coursehero.com/file/249824215/22262q2solpdf/