In the Name of God
Home Work 1– 1392/-/Name:
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1. Show that
a) If P(A) = P(B) = P(AB), then P(A𝐵̅ ⋃ 𝐵 𝐴̅) = 0.
b) If P(A) = P(B) = 1, then P(AB) = 1.
2. Show that P(AB|C) = P(A|BC)P(B|C) and P(ABC) = P(A|BC)P(B|C)P(C).
3. We select at random m objects from a set S of n objects and we denote by 𝐴𝑚
the set of the selected objects. Show that the probability p that a particular
𝑚
element e of S is in 𝐴𝑚 equals 𝑛 .
4. Ten passengers get into a train that has three cars. Assuming a random
placement of passengers, what is the probability that the first car will contain
three of them?
5. A box contains m white and n black balls. Suppose k balls are drawn. Find the
probability of drawing at least one white ball.
6. Your neighbor has 2 children. You learn that he has a son, A. What is the
probability that A’s sibling is a brother?
7. Suppose that five good fuses and two defective ones have been mixed up. To
find the defective fuses, we test them one-by-one, at random and without
replacement. What is the probability that we are lucky and find both of the
defective fuses in the first two tests?
8. Urn 1 contains 5 white balls and 7 black balls. Urn 2 contains 3 whites and 12
black. A fair coin is flipped; if it is Heads, a ball is drawn from Urn 1, and if it
is Tails, a ball is drawn from Urn 2. Suppose that this experiment is done and
you learn that a white ball was selected. What is the probability that this ball
was in fact taken from Urn 2? (i.e., that the coin flip was Tails).
9. Consider independent trials consisting of rolling a pair of fair dice, over and
over. What is the probability that a sum of 5 appears before a sum of 7?
10. Let A and B be independent events with P(A) = 1/4 and P(A ∪ B) = 2P(B)
−P(A). Find
a) P(B);
b) P(A|B);
c) P(𝐵̅|A)