Independent solution

How to solve this Order Statistics question

Setup

Setup

For one observation on the stated interval, the probability of falling below seven is the fraction of the interval below that threshold.

Pr⁡(Xi<7)=710=0.7\Pr(X_i<7)=\frac7{10}=0.7

Model

Model

The sample maximum is below seven exactly when every selected observation is below seven.

{max⁡(X1,X2,X3)<7}=⋂i=13{Xi<7}\{\max(X_1,X_2,X_3)<7\}=\bigcap_{i=1}^{3}\{X_i<7\}

Compute

Compute

Use independence to multiply the three identical probabilities.

Pr⁡(max⁡(X1,X2,X3)<7)=(0.7)3=0.343\Pr(\max(X_1,X_2,X_3)<7)=(0.7)^3=0.343

Answer

Answer

The probability that the maximum is below seven is 0.343.

0.343(D)\boxed{0.343\quad\text{(D)}}