Help! Probability Word Problem.

Question

On a game show, you are given six different digits to arrange in the proper order to represent the price of a house. If your are correct, you win the house. Find the probability of winning under each condition.

 

A) You must guess the position of each digit.

B) You must know the first digit, but must guess the remaining five.

C) You must know the first and last digits, but must guess the remaining four.

Answer 1

 

There are 10 digits .... 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

 

PART A:

 

   For the first digit ... we will remove 0 as an option, since the price has 6 digits (the first one cannot be 0).

 

   For 1st digit:  we have 9 possibilities

   For 2nd digit: we have 9 possibilities ... the 1st digit is removed and 0 is added to the list

   For 3rd digit: we have 8 possibilities ... the 2nd digit is removed from the list of available digits

   For 4th digit: we have 7 possibilities ... the 3rd digit is removed from the list of available digits

   For 5th digit: we have 6 possibilities ... the 3rd digit is removed from the list of available digits

   For 6th digit: we have 5 possibilities ... the 3rd digit is removed from the list of available digits

 

   So the probability is  1/(9•9•8•7•6•5) = 1/136,080

 

PART B:

 

   For first digit:  Given ... so you will get this one right

   For 2nd digit: we have 9 possibilities ... the 1st digit is removed from the list of available digits
   For 3rd digit: we have 8 possibilities ... the 2nd digit is removed from the list of available digits
   For 4th digit: we have 7 possibilities ... the 3rd digit is removed from the list of available digits
   For 5th digit: we have 6 possibilities ... the 3rd digit is removed from the list of available digits
   For 6th digit: we have 5 possibilities ... the 3rd digit is removed from the list of available digits

 

   So the probability is 1/(1•9•8•7•6•5) = 1/15,120

 

PART C:

 

   For first and last digit: Given ... so you will get those 2 right
   For 2nd digit: we have 8 possibilities ... the 1st and last digit is removed from the list of available digits
   For 3rd digit: we have 7 possibilities ... 1 less removed from the list of available digits
   For 4th digit: we have 6 possibilities ... 1 less removed from the list of available digits
   For 5th digit: we have 5 possibilities ... 1 less removed from the list of available digits
   For 6th digit: Given ... so you will get this one right

So the probability is 1/(1•8•7•6•5•1) = 1/1680

 

 

   

 

Answer 2

A) 

The number of possible combinations is 6!, or 720.

Probability of winning is 1/720.

B)

What do you mean by "must know the first digit?"

C)

What do you mean by "must know the first and last digit?"