julie and brandon cut lawns for their neighbors to earn money to a local charity. they thought that the customers should pay 20$. however, sevral customers offered a different pay plan and asked the students to chose. julie and brandon have to decide which of the pay plans would give them a fair deal over the long run. the customer rolls a pair of number cubes. if the product is 24 julie and brandon get 40$. otherwise, they get 10$.

The only way to obtain a product of 24 on 2 dice is to get a 6,4 or 4,6

prob of product of 24 = 2/36 = 1/18

expected value of that = (1/18)(40) = $2.22
expecte value of not a product of 24 = (17/18)(10) = $9.44

strange question!
where does "over the long run" come in?
What happened to the $20 charge?

thanks

To determine which pay plan would give Julie and Brandon a fair deal over the long run, we need to calculate the expected value for each pay plan.

Pay Plan 1: $40 if the product of the numbers rolled is 24, otherwise $10.

To calculate the expected value for Pay Plan 1, we need to consider all possible outcomes for rolling two number cubes and find the probability of each outcome:

Possible outcomes for rolling two number cubes:

1. Product of 24: (3, 8), (8, 3), (4, 6), (6, 4) - 4 outcomes
2. All other products: (1, 1), (1, 2), (1, 3), ..., (6, 6), excluding the ones from the first category - 36 - 4 = 32 outcomes

The probability of getting the product of 24 is 4/36 = 1/9, and the probability of getting any other product is 32/36 = 8/9.

Expected value for Pay Plan 1:
(40 * 1/9) + (10 * 8/9) = 40/9 + 80/9 = 120/9 = $13.33 (rounded to two decimal places)

Pay Plan 2: $20 regardless of the outcome.

Expected value for Pay Plan 2:
$20, since it remains constant regardless of the outcome.

Comparing the expected values for both pay plans, Pay Plan 1 has an expected value of $13.33, while Pay Plan 2 has an expected value of $20.

Therefore, to ensure a fair deal over the long run, Julie and Brandon should choose Pay Plan 2 where they earn a constant $20 per customer, regardless of the outcome of rolling the number cubes.