Independent Events and Conditional Probability Questions Response

Problem set 1

please answer all 8 questions with all the required steps shown

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Problem set 1
May 29, 2020
Problem 1
If A, B and D are three events such that P (A ∪ B ∪ D) = 0.7, what is the value of
P (AC ∩ B C ∩ DC ) ?
Problem 2
Prove the following identity for any three events A, B and C:
P (A ∪ B ∪ D) = P (A) + P (B) + P (C) − P (A ∩ B) − P (A ∩ C) − P (B ∩ C) + P (A ∩ B ∩ C)
Problem 3
Suppose that the events A and B are disjoint. Under what conditions are AC and B C
disjoint?
Problem 4
Suppose that a committee of 12 people is selected in a random manner from a group of 100
people. Determine the probability that two particular people A and B will both be selected.
Problem 5
Consider an experiment in which a fair coin is tossed once and a balanced die is rolled once.
• Describe the sample space for this experiment.
• What is the probability that a head will be obtained on the coin and an odd number
will be obtained on the die?
Problem 6
Assuming that A and B are independent events, prove that the events AC and B C are also
independent.
1
Problem set 1
May 29, 2020
Problem 7
Suppose that A, B, and C are three independent events such that P (A) = 1/4, P (B) = 1/3,
and P (C) = 1/2.
• Determine the probability that none of these three events will occur.
• Determine the probability that exactly one of these three events will occur.
Problem 8
A box contains five coins with a head on each side, three coins with a tail on each side, and
three fair coins. If one of these eleven coins is selected at random and tossed once, what is
the probability that a head will be obtained?
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