IE 227 RECITATION - 2
02.10.2013
2.21 Consider the experiment of tossing a coin once and observing whether it comes up a head
(H) or tail (T). The sample space for this random experiment is Ω={H,T}. Let p be a real number
with 0 ≤ 𝑝 ≤ 1.
a) Explain why the probability assignment 𝑃 = ({𝐻}) = 𝑝 and 𝑃 = ({𝑇}) = 1 − 𝑝determines a
unique probability measure on Ω.
b) Based on the probability assignment in part (a), find the probability of each of the four
events of this random experiment.
Answer: a)Unique probability measure corresponds to:
1) p≥0
2) p+(1-p)=1 or p≤1
b) …
2.24 Let Ω be the sample space for a random experiment and let P be a probability measure on
Ω. Use the Kolmogorov axioms to verify the following for events A and B.
a) 𝑃(𝐵) = 𝑃(𝐵 ∩ 𝐴) + 𝑃(𝐵 ∩ 𝐴𝑐 )
b) 𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵 ∩ 𝐴𝑐 )
Answer: Ω: Sample Space
Kolmogorov Axioms: 1) P(E)≥0, 2)P(Ω)=1, 3)𝑃(⋃𝑛 𝐴𝑛 ) = ∑𝑛 𝑃(𝐴𝑛 ) if 𝐴𝑛 s are M.E.
a) a) 𝑃(𝐵) = 𝑃(𝐵 ∩ 𝐴) + 𝑃(𝐵 ∩ 𝐴𝑐 )
A
B
BcA AB AcB
…
2.42 A commuter train arrives punctually at a station every half hour. Each morning, a commuter
named John leaves his house and casually strolls to the train station. Finf the probability that
John waits for the train
a) between 10 and 15 minutes.
b) At least 10 minutes.
Answer: a) A: event that commuter named John waits for the train between 10 and 15 minutes.
A={x: 10≤x≤15, x€R} x:wait time
Ω: 30 minutes
P(A)=P(10≤x≤15)
𝑛(10≤x≤15)
=
𝑛(Ω)
…
2.65 According to Current Population Reports, published by the U.S. Bereau of the Census, 51%
of U.S adults are female, 7.1% are divorced, and 4.1% are divorced females. Dtermine the
probability that a U.S. adult selected at random is
a) Either female or divorced
b) A male
c) A female but not divorced
d) A divorced male
Answer: F: adult selected is female, P(F)=0,51
D: adult selected is divorced
F
D
b
a
c
d
…
2.85 Complete the following table.
Event
Description
𝐴 ∩ 𝐵𝐶
A occurs but B doesn’t.
𝐴 ∪ 𝐵𝐶
Neither A nor B occurs
(𝐴 ∩ 𝐵𝐶 ) ∪ (𝐴𝐶 ∩ 𝐵)
Events A, B and C occur
𝐴 ∩ (𝐵 ∪ 𝐶)
Either A occurs or both B and C occur
𝐴∪𝐵∪𝐶
At least one of A, B and C does not occur
⋂ 𝐴𝑛
𝑛
At least one of 𝐴1 , 𝐴2 , … occur