Question

I. At a certain school, 60% of the students wear neither a ring nor a necklace, 20% wear a ring, 30% wear a necklace. Compute the probability that a randomly selected student wears (a) a ring or a necklace; (b) a ring and a necklace 2. A school offers three language classes: Spanish (S), French (F), and German (G). There are 100 students total, of which 28 take S. 26 take F, 16 take G, 12 take both S and F, 4 take both S and G, 6 take both F and G, and 2 take all three languages. (1) Compute the probability that a randomly selected student (a) is not taking any of the three language classes; (b) takes exactly one of the three language classes. (2) Compute the probability that, of two randomly selected students, at least one takes a language class. 3. A woman has n keys, of which one will open her door. (a) If she tries the keys at random, discarding those that do not work,what is the probability that she will open the door on her kth try? (b) What if she does not discard previously tried keys? 4. Two fair dice are rolled. Compute the probability that the number on the first is smaller than the number on the second. 5. If 4 married couples are arranged in a row, find the probability that no husband sits next to his wife. Hint. Let Ak"kth couple sits together", k 1,2,3,4.Apply inclusion/exclusion principle and think about the book arrangement on the shelf P (at least one couple sits together) i<j<k For instance with i < j. AiAj - "the ith couple and the jth couple go together, the remaining 4 people are arranged in any order".

Answer 1

1)

P(neither ring nor necklace) = 0.60

P(ring)=0.20

P(necklace)=0.30

let P(ring and necklace) = x

total sample space = 1

P(ring or necklace)+P(neither ring nor necklace) = 1

P(ring)+P(necklace)- P(ring and necklace) +P(neither ring nor necklace) =1

0.20+0.30-x+0.60=1

x = 0.10

so, P(ring and necklace) =0.10

a) P(necklace or ring) = P(ring)+P(necklace)- P(ring and necklace)=0.20+0.30-0.10=0.40

b)P(ring and necklace) = 0.10 (as calculated earlier)

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