Question

Ten playing cards numbered \(1,\ldots,10\) are drawn with replacement. What is the probability that the second number is greater than the first (rounded to two decimal places)?

Hint:

For two i.i.d. discrete draws, \(P(Y>X)=\dfrac{1-P(Y=X)}{2}\).

Answer 1

Correct Answer: 0.45

Step 1: With replacement, the two draws are i.i.d. uniform on \(\{1,\ldots,10\}\).
Step 2: By symmetry, \(P(Y>X)=P(Y<X)\). Also \(P(Y=X)=\frac{10}{10\cdot10}=0.10\).
Step 3: Hence \(2P(Y>X)=1-0.10\Rightarrow P(Y>X)=0.45\).
So the required probability is \(\boxed{0.45}\).