IXL - Choose the better bet (Precalculus practice)

There are two different raffles you can enter.

Raffle A, which is for a fundraiser, has 250 tickets. Each ticket costs $13. One ticket will win a $450 prize, and the remaining tickets will win nothing.

In raffle B, one ticket out of 125 will win a $1,000 prize. The rest will win nothing. Each ticket costs $19.

Which raffle is a better deal?

Raffle A

Raffle B

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Learn with an example

question

There are two different raffles you can enter.

Raffle A, which is for a fundraiser, has 50 tickets. Each ticket costs $10. One ticket will win a $290 prize, and the remaining tickets will win nothing.

In raffle B, one ticket will win a $680 prize, one ticket will win a $510 prize, one ticket will win a $330 prize, one ticket will win a $250 prize, and the rest will win nothing. There are 500 in the raffle, each costing $7.

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 $10 to enter the raffle, the cost is $10.

Find the expected payoff.

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

Since the only prize is $290, the values

can take are $290 and $0 (if no prize is won).

Since 1 ticket has a payoff of $290, the probability of winning $290 is

0.02

That is,

290)

0.02

Since the 49 remaining tickets have a payoff of $0, the probability of winning $0 is

0.98

That is,

0.98

Write the probability distribution of

in the table.

X	P ( X )
$0	0.98
$290	0.02

To find

(

)

the expected value of

find

(

)

for each value

that

can take. Then, sum these terms.

X	P ( X )	X P ( X )
$0	0.98	$0
$290	0.02	$5.80

(

)

5.8

$5.80

The expected payoff is $5.80.

Find the expected profit.

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

$5.80

–

$10

-$4.20

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 $7 to enter the raffle, the cost is $7.

Find the expected payoff.

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

Since the prizes are $680, $510, $330, and $250, the values

can take are $680, $510, $330, $250, and $0 (if no prize is won).

Since 1 ticket has a payoff of $680, the probability of winning $680 is

500

0.002

That is,

680)

0.002

Since 1 ticket has a payoff of $510, the probability of winning $510 is

500

0.002

That is,

510)

0.002

Since 1 ticket has a payoff of $330, the probability of winning $330 is

500

0.002

That is,

330)

0.002

Since 1 ticket has a payoff of $250, the probability of winning $250 is

500

0.002

That is,

250)

0.002

Since exactly 4 tickets win a prize, the 496 remaining tickets have a payoff of $0. So, the probability of winning $0 is

496

500

0.992

That is,

0.992

Write the probability distribution of

in the table.

Y	P ( Y )
$0	0.992
$250	0.002
$330	0.002
$510	0.002
$680	0.002

To find

(

)

the expected value of

find

(

)

for each value

that

can take. Then, sum these terms.

Y	P ( Y )	Y P ( Y )
$0	0.992	$0
$250	0.002	$0.50
$330	0.002	$0.66
$510	0.002	$1.02
$680	0.002	$1.36

(

)

0.5

0.66

1.02

1.36

$3.54

The expected payoff is $3.54.

Find the expected profit.

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

$3.54

–

-$3.46

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

Since

-$4.20

-$3.46

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

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