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In one of the islands that his travels took him to, Gulliver noticed that the probability that a (uniformly) randomly chosen inhabitant has height at least $2$ meters is $0.2$. Also, $0.2$ is the probability that a a (uniformly) randomly chosen inhabitant has height at most $1.5$ meters. What can we conclude about the average height $h$ in meters of the inhabitants of the island?

  1. $1.5 \leq h \leq 2$
  2. $h \geq 1.3$
  3. $h \leq 2.2$

Which of the above statements is necessarily true?

  1. ii only
  2. iii only
  3. i, ii and iii
  4. ii and iii only
  5. None of the above
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we can approach the question in this way:

say there are 100 inhabitants

1.) 20 inhabitants have height atleast 2 m.

2.) 20 inhabitants have height atmost 1.5m

3.) 60 people have height between 1.5m and 2m

Consider the lower limit of height in case 1. It is 2m.(upper limit is infinity)

In case 2 the lower limit can be 0.(upper limit is 1.5m)

In case 3 the lower limit is 1.5m(upper limit 2m)

On taking the average of minimum values:(as the options are comparing values of heights <= or >=)

2 x 20+0 x 20+1.5 x 60 = 130

So lower limit of height is : h>=1.3

So answer is option A
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