Question

In a simple lottery, 10 ping pong balls numbered 1 through 10 are placed in a bucket and mixed thoroughly. Four balls are selected at random without replacement and their number is recorded. To play, you purchase a ticket for $1 and write down a 4-digit number. If the numbers match your ticket, you win $5,000.

a) How many different outcomes are possible?

b) What is your probability of winning.

c) Suppose that you can purchase a Mega ticket for $20. This means that the order does not matter for you to win. How many outcomes are now possible?

d) What is your probability of winning with a Mega ticket?

e) If you had $20 to play this lottery, would you rather buy 20 regular tickets or 1 Mega ticket?

Answer 1

a) How many different outcomes are possible?

Here we have 10 balls and we have to choose 4 without replacement means numbers cannot be repeated.. So we can use permutation or combination function.

But since the order of the digit is important we will use permutation function. Order is important means

1234 is not same as 2314.

So the number of possible combinations = nPr

=

So we have 4 to choose from 10

Ways = 10P4

Using the formula

Possible outcomes = 5040

b) What is your probability of winning.

There is only oe winner. So there is only one winning order or a combination.

There out of 5040 only 1 is winning

Probabiltiy= 1 / 5040

Probabiity = 0.000198

c) Suppose that you can purchase a Mega ticket for $20. This means that the order does not matter for you to win. How many outcomes are now possible?

Here order is not important so we can use combination function.

Means 1234 is not same as 2314.

Formula = nCr = = (nPr) / r!

10C4 = 5040 / (4!)

Possible outcomes = 210

d) What is your probability of winning with a Mega ticket?

Again there is only one way possible that same numbered balls are picked as matched with the written one.

S probability = 1/ 210

Probability = 0.00477

e) If you had $20 to play this lottery, would you rather buy 20 regular tickets or 1 Mega ticket?

Here we can decide by see our net expected winning

Net winning= Profit - bet amount

= Profit - 20

Here we know that we are going to certainly lose $20 to buy the ticket so the winning of $5000 will decide the option to choose.

Here if we buy 1 mega ticket the probabiltiy = 0.00477

So the expected winning = 5000 * 0.00477

1 mega ticket = $23.809

If we are buying 20 regular then we can 20 chances out of 5040 probabiltiy = 20 / 5040 = 0.00397

So the expected = 0.00397 * 5000

20 regular ticket = $19.84

Since by buying 1 mega ticket the stakes are lower and expected amount is more we should

buy 1 mega ticket.