A point drawn from the unit interval

Topic: Random experiments and probability · more questions on this topic
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A point is drawn from [0,1] so that probabilities are lengths.

  1. Find the probability that the point is less than 1/2.

  2. Find the probability that it lies between 1/3 and 3/4.

  3. The draw is repeated independently. Find the probability that the sum of the two points exceeds 1/2.

  1. The length of [0,1/2) is 1/2.

  2. The length of [1/3,3/4] is 3/4-1/3=5/12.

  3. The two draws are uniform on the unit square. The corner \{(x,y):x\ge 0,\,y\ge 0,\,x+y\le 1/2\} is a right triangle of area (1/2)\times(1/2)\times(1/2)=1/8. The complementary probability is 1-1/8=7/8.