Answer to Question #126845 in Statistics and Probability for Ruth Lucy

Question

Suppose John writes down a single number that is in the range from 1 to 20, and Susan also writes down a single number in the range from 1 to 20. What is the probability that the number John wrote down will be higher than Susan’s?

Answer 1

Solution:

Let A_J be the event that the number John wrote down will be higher than Susan’s.

Let A_S be the event that the number Susan wrote down will be higher than John’s.

Let Ae be the event that the number John wrote down will be equal the number Susan wrote down.

Then

$A_J \bigcup A_S \bigcup A_e = U$

$A_J \bigcup A_S=\varnothing$

$A_J \bigcup A_e=\varnothing$

$A_S \bigcup A_e=\varnothing$

And P(A_J)+P(A_S)+P(A_e)=1.

P(A_J)=P(A_S) for reasons of symmetry.

So 2P(A_J)+P(A_e)=1,

P(A_J)=(1-P(A_e))/2,

P(A_e)=m/n.

There are 20²=400 ways to position that John writes down a single number that is in the range from 1 to 20, and Susan also writes down a single number in the range from 1 to 20.

So n=20²=400.

There are 20 ways to position that the number John wrote down will be equal the number Susan wrote down:

(i,i), i=1,2,...,20

So m=20.

P(A_e)=20/400=1/20.

P(A_J)=(1-1/20)/2=19/40.

Answer:

The probability that the number John wrote down will be higher than Susan’s is equal 19/40.