Question

1. Students in probability classes sometimes wonder by probabilists seem to have a peculiar fascination with balls being thrown at random into boxes or drawn at random from urns. It's not just because probabilists are weird. It's because the image of balls and boxes is one unified way of thinking about repeated trials in diverse settings. For example, balls thrown at random into boxes independently of each other is a way of visualizing independent repeated trials that have equally likely outcomes. Tossing a coin 10 times and keeping track of heads and tails is like throwing 10 balls independently at random into two boxes labeled H and T. The number of heads is the number of balls that fall in Box H. Rolling a die 15 times and keeping track of the faces that appear is like throwing 15 balls independently at random into six boxes labeled 1,2,3,4,5,6 . Sampling ? times at random with replacement from a population of ? individuals is like throwing ? balls independently at random into ? boxes labeled 1,2,…,? . a) A random number generator that draws digits at random with replacement from {0,1,2,3,4,5,6,7,8,9} is run three times. Find the chance that the digits drawn can form the number 105 by rearrangement if necessary. Start by filling in the blanks in the following sentence; the first two blanks should be filled with integers. "This experiment is like throwing ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ balls independently at random into ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ boxes labeled ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ ." Then imagine where the balls must land to make the event occur. b) Five people enter an Evans Hall elevator (going up) on the first floor. Assume that each person presses the button for one of floors 2 through 10 at random, independently of all others. Find the chance that they press five different buttons. Start by filling in the blanks in the following sentence; the first two blanks should be filled with integers. "This experiment is like throwing ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ balls independently at random into ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ boxes labeled ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ ." Then imagine where the balls must land to make the event occur.

Answer 1

Solution:

a) A random number generator that draws digits at random with replacement from {0,1,2,3,4,5,6,7,8,9} is run three times. Find the chance that the digits drawn can form the number 105 by rearrangement if necessary.

This experiment is like throwing 3 balls independently at random into 10 boxes labeled 0,1,2,3,4,5,6,7,8,9.

b)Five people enter an Evans Hall elevator (going up) on the first floor. Assume that each person presses the button for one of floors 2 through 10 at random, independently of all others. Find the chance that they press five different buttons.

This experiment is like throwing 5 balls independently at random into 9 boxes labeled 2,3,4,5,6,7,8,9.