Question

assignment #7 - study guide

1. of the two hundred works of art at a large gallery, 118 are paintings, and 166 are for sale. when a work

of art is selected at random, let the event that it is a painting be a and the event that it is for sale be b.
a. what are the values of p(a) and p(b)?
b. suppose you are told that p(a and b) = 0.51. find p(a ∪ b). explain what p(a ∪ b) means in the
context of this problem.
c. suppose now that you are not given the information in part (b), but you are told that the events a
and b are independent. find p(a or b).
d. find p(a ∩ b) assuming events a and b are independent. explain what p(a ∩ b) means in the
context of this problem.

Explanation:

Step1: Calculate $P(A)$

$P(A) = \frac{\text{Number of paintings}}{\text{Total art works}} = \frac{118}{200} = 0.59$

Step2: Calculate $P(B)$

$P(B) = \frac{\text{Art works for sale}}{\text{Total art works}} = \frac{166}{200} = 0.83$
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Step3: Calculate $P(A \cup B)$ (part b)

Use addition rule: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$
$P(A \cup B) = 0.59 + 0.83 - 0.51 = 0.91$

Step4: Interpret $P(A \cup B)$

This is the probability a randomly selected work is a painting, for sale, or both.
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Step5: Find $P(A \cap B)$ (independent events, part c)

For independent events: $P(A \cap B) = P(A) \times P(B)$
$P(A \cap B) = 0.59 \times 0.83 = 0.4897$

Step6: Calculate $P(A \cup B)$ (part c)

$P(A \cup B) = 0.59 + 0.83 - 0.4897 = 0.9303$
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Step7: Calculate $P(A')$ and $P(B')$ (part d)

$P(A') = 1 - P(A) = 1 - 0.59 = 0.41$
$P(B') = 1 - P(B) = 1 - 0.83 = 0.17$

Step8: Calculate $P(A' \cap B')$ (independent complements)

For independent events, complements are independent: $P(A' \cap B') = P(A') \times P(B')$
$P(A' \cap B') = 0.41 \times 0.17 = 0.0697$

Step9: Interpret $P(A' \cap B')$

This is the probability a randomly selected work is not a painting and not for sale.

Answer:

a. $P(A) = 0.59$, $P(B) = 0.83$
b. $P(A \cup B) = 0.91$; this is the probability a randomly selected work is a painting, for sale, or both.
c. $P(A \text{ or } B) = 0.9303$
d. $P(A' \cap B') = 0.0697$; this is the probability a randomly selected work is neither a painting nor for sale.