Question

An urn contains 3 ​one-dollar bills, 1​ five-dollar bill and 1​ ten-dollar bill. A player draws bills one at a time without replacement from the urn until a​ ten-dollar bill is drawn. Then the game stops. All bills are kept by the player.​ Determine: ​(A)  The probability of winning ​$11 . ​(B)  The probability of winning all bills in the urn. ​(C)  The probability of the game stopping at the second draw.

Answer 1

The total number of bills = 3 + 1 + 1 = 5

The probability of choosing a particular bill = No. of particular bill / Total bills.

Also for every next draw the number of bills decreases.

​(A)  The probability of winning ​$11 .

The games stops after drawing $10 bill.

So if we are winning means the last draw was $10. Here we win $11, that means there is more than 1 draw.

There can be 2 draws only of $1 and $10.

The probability of getting $1 = 3 / 5 .........there are 3 $1 bills to choose from and a total of 5 bills

The probability of getting $10 = 1 / 4 .........there is one $10 bill to choose from and a total of 4 bills remaining.

Probability = (3/5) * (1 / 4)

ANS: 0.15

​(B)  The probability of winning all bills in the urn.

Winning all the bills. There can be 5 draws in all with $10 bill being last 1.

The drawing of remaining 4 bills can be in any order. So each time a bill is drawn the number of bills apart from $10 will reduce and so will the total number. The last will have 1 option of total as well as choice of $10.

Prob = (4/5) * (3/4) * (2/3) * (1/2) * (1/1)

ANS: 0.2

It can be thought of this way as well. We can have 5 draws at the most. To win all bills, we need to have all five draws. There is only 1 way to do that if we choose $10 last. So out of 5, we have 1 to win all. Therefore 1/5 = 0.2.

​(C)  The probability of the game stopping at the second draw.

The game stops at second draw means 2nd draw is $10. So the first can be either one $1 bill or one $5 bill.

Prob of 1st draw being $1 = 3 /5

Prob of 1st draw being $5 = 1 /5

Combined probability of 1st draw not being $10 = 3/5 + 1/5

= 4/5

Prob of 2nd draw being $10= 1 / 4 ..........since 1 bill is already chosen

Prob = 1st draw * 2nd draw

= 4/5 * 1/4

ANS: 0.2