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# Among all the students at a certain high school, the probability of pi

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Math Expert
Joined: 02 Sep 2009
Posts: 61549
Among all the students at a certain high school, the probability of pi  [#permalink]

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10 Dec 2019, 01:04
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95% (hard)

Question Stats:

45% (02:38) correct 55% (02:21) wrong based on 47 sessions

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Among all the students at a certain high school, the probability of picking a left-handed student is $$\frac{1}{4}$$, and the probability of picking a student who is learning Spanish is $$\frac{2}{3}$$ Which of the following could be the probability of picking a student who is either left-handed or learning Spanish or both?

I. $$\frac{2}{3}$$
II. $$\frac{3}{4}$$
III. $$\frac{5}{6}$$

A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III

Are You Up For the Challenge: 700 Level Questions

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Joined: 19 Oct 2018
Posts: 1314
Location: India
Re: Among all the students at a certain high school, the probability of pi  [#permalink]

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15 Dec 2019, 00:51
2
Assume total students = LCM(3,4)=12

$$N(l)= \frac{1}{4}*12=3$$
$$N(s)= \frac{2}{3}*12=8$$

We need the minimum and maximum value of [N(l) U N(s)]

[N(l) U N(s)]= N(l) + N(s) - [N(l) ∩ N(s)]

1. For the maximum value, minimize [N(l) ∩ N(s)].
Minimum possible value of [N(l) ∩ N(s)]=0

[N(l) U N(s)]= N(l) + N(s)-0
[N(l) U N(s)]= 3+8=11

2. For the minimum value, maximize [N(l) ∩ N(s)].
Maximum possible value of [N(l) ∩ N(s)]=3

[N(l) U N(s)]= N(l) + N(s)-3
[N(l) U N(s)]= 3+8-3=8

probability of picking a student who is either left-handed or learning Spanish or both = P

$$\frac{8}{12} ≤ P ≤ \frac{11}{12}$$

All 3 options lie in the range.

Bunuel wrote:
Among all the students at a certain high school, the probability of picking a left-handed student is $$\frac{1}{4}$$, and the probability of picking a student who is learning Spanish is $$\frac{2}{3}$$ Which of the following could be the probability of picking a student who is either left-handed or learning Spanish or both?

I. $$\frac{2}{3}$$
II. $$\frac{3}{4}$$
III. $$\frac{5}{6}$$

A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III

Are You Up For the Challenge: 700 Level Questions
Math Expert
Joined: 02 Aug 2009
Posts: 8254
Among all the students at a certain high school, the probability of pi  [#permalink]

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10 Dec 2019, 06:56
Bunuel wrote:
Among all the students at a certain high school, the probability of picking a left-handed student is $$\frac{1}{4}$$, and the probability of picking a student who is learning Spanish is $$\frac{2}{3}$$ Which of the following could be the probability of picking a student who is either left-handed or learning Spanish or both?

I. $$\frac{2}{3}$$
II. $$\frac{3}{4}$$
III. $$\frac{5}{6}$$

A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III

Are You Up For the Challenge: 700 Level Questions

Use 2 by 2 matrix
Let the total number be 120 as the fractions have denominator 3 and 4
............S........NS
L...........b....... a.....$$120*\frac{1}{4}=30$$
R..........c.........d.... $$90$$
Total.....80......40....$$120$$

Now we are looking at
$$\frac{(a+b+c)}{120}$$
Least value will be when a=0 and b=30 and c=50....$$\frac{80}{120}=\frac{2}{3}$$
Greatest value will be when b=0 and a=30 and c=80...$$\frac{(80+30)}{120}=\frac{11}{12}$$

All the fractions are within this range, so E

Let us test

I. $$\frac{2}{3}$$....b=30, a=0 and c=50
II. $$\frac{3}{4}$$.... b=20, a=10 and c=60
III. $$\frac{5}{6}$$...b=10, a=20 and c=70
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Joined: 24 Nov 2019
Posts: 6
Among all the students at a certain high school, the probability of pi  [#permalink]

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14 Dec 2019, 23:28
[quote="Bunuel"]Among all the students at a certain high school, the probability of picking a left-handed student is $$\frac{1}{4}$$, and the probability of picking a student who is learning Spanish is $$\frac{2}{3}$$ Which of the following could be the probability of picking a student who is either left-handed or learning Spanish or both?

I. $$\frac{2}{3}$$
II. $$\frac{3}{4}$$
III. $$\frac{5}{6}$$

A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III

Hi Bunuel,

Could you please share the solution?
Among all the students at a certain high school, the probability of pi   [#permalink] 14 Dec 2019, 23:28
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