If we select 20 numbers at random from the interval{0,1} a) Find the probability

Question

If we select 20 numbers at random from the interval{0,1} a) Find the probability that the first 10 numbers are lessthan 1/2 and the last 10 numbers are grater than 3/4 b) Find the probability that the first 5 number are lessthan 0.2, the next 10 numbers are between 0.2 and 0.8 and the last5 numbers are more than 0.8 If we select 20 numbers at random from the interval{0,1} a) Find the probability that the first 10 numbers are lessthan 1/2 and the last 10 numbers are grater than 3/4 b) Find the probability that the first 5 number are lessthan 0.2, the next 10 numbers are between 0.2 and 0.8 and the last5 numbers are more than 0.8

Explanation / Answer

Let U be uniform over (0,1), then
(a) p1=P(U<.5)=.5 p2=P(U>.75)=.25 We have P(first10<.5)=p110 =.510=.000977    and       P(2nd10>.75)=p210=.2510=9.54x10-7 Hence the probability the first 10 are <1/2 and the secondten are >3/4 is .000977(9.54x10-7)=9.317x10-10=0 (b) P(U<.2)=.2 P(.2<U<.8)=.6 P(U>.8)=.2 Therefore the desired probability is .25(.6)10(.2)5=6.1917x10-10=0 Hence the probability the first 10 are <1/2 and the secondten are >3/4 is .000977(9.54x10-7)=9.317x10-10=0 (b) P(U<.2)=.2 P(.2<U<.8)=.6 P(U>.8)=.2 Therefore the desired probability is .25(.6)10(.2)5=6.1917x10-10=0

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