IXL - Choose the better bet (Algebra 2 practice)

There are two different raffles you can enter.

Raffle A, which is at a carnival, has 125 tickets. Each ticket costs $8. One ticket will win a $400 prize, and the remaining tickets will win nothing.

Raffle B has 50 tickets. Each ticket costs $4. One ticket will win a $270 prize, one ticket will win a $250 prize, one ticket will win a $140 prize, one ticket will win a $50 prize, and the remaining tickets will win nothing.

Which raffle is a better deal?

Raffle A

Raffle B

ref_doc_title.

Learn with an example

question

There are two different raffles you can enter.

In raffle A, one ticket will win a $750 prize, and the rest will win nothing. There are 100 in the raffle, each costing $14.

Raffle B has 50 tickets. Each ticket costs $4. One ticket will win a $720 prize, and the remaining tickets will win nothing.

Which raffle is a better deal?

Raffle A

Raffle B

key idea

A random variable is a function that assigns a numerical value to each outcome in a sample space. Note that though random variables are functions, they are usually denoted by capital letters like

is a random variable,

(

)

is the probability that

takes the value

The function that maps

(

)

is the probability distribution of

A probability distribution is frequently represented as an equation, table, or graph.

The expected value or mean

(

)

of a random variable

is the sum of the terms

(

)

for each value

that

can take. It is also called the expected value or mean of the probability distribution of

and is sometimes written

𝜇

solution

Find the raffle with the greater expected profit.

start of large bold text, Calculate the expected profit of raffle A., end of large bold text

The expected profit is the expected payoff minus the cost. Since it costs $14 to enter the raffle, the cost is $14.

Find the expected payoff.

Use the random variable whose value is the payoff in the raffle. Call it

Since the only prize is $750, the values

can take are $750 and $0 (if no prize is won).

Since 1 ticket has a payoff of $750, the probability of winning $750 is

100

0.01

That is,

750)

0.01

Since the 99 remaining tickets have a payoff of $0, the probability of winning $0 is

100

0.99

That is,

0.99

Write the probability distribution of

in the table.

X	P ( X )
$0	0.99
$750	0.01

To find

(

)

the expected value of

find

(

)

for each value

that

can take. Then, sum these terms.

X	P ( X )	X P ( X )
$0	0.99	$0
$750	0.01	$7.50

(

)

7.5

$7.50

The expected payoff is $7.50.

Find the expected profit.

The expected profit of raffle A is the expected payoff minus the cost:

$7.50

–

$14

-$6.50

start of large bold text, Calculate the expected profit of raffle B., end of large bold text

The expected profit is the expected payoff minus the cost. Since it costs $4 to enter the raffle, the cost is $4.

Find the expected payoff.

Use the random variable whose value is the payoff in the raffle. Call it

Since the prizes are $720, the values

can take are $720, and $0 (if no prize is won).

Since 1 ticket has a payoff of $720, the probability of winning $720 is

0.02

That is,

720)

0.02

Since exactly

–

949

tickets win a prize, the 999 remaining tickets have a payoff of $0. So, the probability of winning $0 is

999

19.98

That is,

19.98

Write the probability distribution of

in the table.

Y	P ( Y )
$0	19.98
$720	0.02

To find

(

)

the expected value of

find

(

)

for each value

that

can take. Then, sum these terms.

Y	P ( Y )	Y P ( Y )
$0	19.98	$0
$720	0.02	$14.40

(

)

14.4

$14.40

The expected payoff is $14.40.

Find the expected profit.

The expected profit of raffle B is the expected payoff minus the cost:

$14.40

–

$10.40

start of large bold text, Choose the raffle with the greater expected profit., end of large bold text

Since

-$6.50

$10.40

raffle B has a greater expected profit. Raffle B is a better deal.

$Confetti$

Excellent!

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