tut01_sol

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Massachusetts Institute of Technology

Department of Electrical Engineering & Computer Science6.041/6.431: Probabilistic Systems Analysis

(Spring 2015)

Tutorial 1

February 6, 2015

1. Venn Diagrams

In this problem, you are given descriptions in words of certain events (e.g.,“at least one of theevents A,B,C occurs”). For each one of these descriptions, identify the correct symbolic descrip-tion in terms of A,B,C from Events E1-E7 below. Also identify the correct description in termsof regions (i.e., subsets of the sample space Ω) as depicted in the Venn diagram below. (Forexample, Region 1 is the part of A outside of B and C.)

Figure 1: VENN DIAGRAMS

Symbolic descriptions:

• Event E1: A ∩B ∩C

• Event E2: (A ∩B ∩C )c

• Event E3: A ∩B ∩C c

• Event E4: B ∪ (Bc ∩ C c)

• Event E5: Ac ∩Bc ∩ C c

• Event E6: (A ∩B) ∪ (A ∩ C ) ∪ (B ∩C )

• Event E7: (A ∩Bc ∩C c) ∪ (Ac ∩B ∩ C c) ∪ (Ac ∩Bc ∩ C )

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Descriptions in words of certain events:

• At least two of the events A, B, C occur.

• At most two of the events A, B, C occur.

• None of the events A, B, C occurs.

• All three events A, B, C occur.

• Exactly one of the events A, B, C occurs.

• Events A and B occur, but C does not occur.

• Either event B occurs or, if not, then C also does not occur.

2. Three Tosses of a Fair Coin

You flip a fair coin (i.e., the probability of obtaining Heads is 1/2) three times. Assume that allsequences of coin flip results, of length 3, are equally likely. Determine the probability of each of the following events.

Questions:

(a) HHH : 3 Heads:

(b) HTH : the sequence Heads, Tails, Heads:

(c) Any sequence with 2 Heads and 1 Tails (in any order):

(d) Any sequence in which the number of Heads is greater than or equal to the number of Tails:

3. Geniuses and Chocolate

Out of the students in a class, 60% are geniuses, 70% love chocolate, and 40% fall into bothcategories. Determine the probability that a randomly selected student is neither a genius nor achocolate lover.

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(e) Exactly one of the events A, B, C occurs:(A ∩Bc ∩C c) ∪ (Ac ∩B ∩C c) ∪ (Ac ∩Bc ∩ C )

(f) Events A and B occur, but C does not occur:

A ∩B ∩C c

(g) Either event B occurs or, if not, then C also does not occur:B ∪ (Bc ∩ C c)

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2. Since all outcomes are equally likely, we are dealing with a discrete uniform probability law. Toobtain the probability of an event, we simply count the number of elements in the event anddivide by the total number of elements in the sample space.

There are 3 flips, with 2 possible results for each flip. Thus there are 23 = 8 elements (distinctsequences) in the sample space.

(a) Any particular sequence has probability 1/8. Therefore, P(HHH ) = 1/8 .

(b) This event again consists of a single sequence, and so P(HTH ) = 1/8 .

(c) The event of interest is HHT,HTH,THH . Since it consists of 3 elements, its probability

is 3/8 .

(d) The set of sequences that have at least as many Heads as Tails is HHH,HHT,HTH,THH .

Its probability is 4/8 .

3. MITr, Unit 1, Solved problem 2.

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