Math  /  Data & Statistics

QuestionYou need to borrow money for gas, so you ask your mother and your sister. You can only borrow money from one of them. Before giving you money, they each say they will make you play a game. Your sister says she wants you to spin a spinner with two outcomes, blue and red, on it. She will give you $2\$ 2 if the spinner lands on blue and $18\$ 18 if the spinner lands on red. Your mother says she wants you to roll a six-sided die. She will give you $2\$ 2 times the number that appears on the die. Determine the expected value of each game and decide which offer you should take.

Studdy Solution

STEP 1

What is this asking? Which game gives me more money on average: spinning a spinner or rolling a die? Watch out! Don't mix up the money amounts with the probabilities!

STEP 2

1. Sister's Spinner Game
2. Mother's Dice Game
3. Compare and Decide

STEP 3

Let's figure out the expected value of the spinner game!
We need to know the probability of each outcome and the money we get for each outcome.

STEP 4

The spinner has two outcomes: blue and red.
Since there are only two outcomes, and assuming they're equally likely, the probability of landing on blue is 12\frac{1}{2}, and the probability of landing on red is also 12\frac{1}{2}.

STEP 5

Now, let's calculate the expected value.
The expected value is the sum of each outcome's value multiplied by its probability.
So, for the spinner, it's $212+$1812\$2 \cdot \frac{1}{2} + \$18 \cdot \frac{1}{2}.

STEP 6

Calculating that out, we get $1+$9=$10\$1 + \$9 = \$10.
So, the expected value of playing the spinner game is $10\$10!

STEP 7

Time for the dice game!
A six-sided die has the numbers 1 through 6, each with a probability of 16\frac{1}{6} of showing up.

STEP 8

Our mom gives us $2\$2 times the number on the die.
So, if we roll a 1, we get $21=$2\$2 \cdot 1 = \$2.
If we roll a 2, we get $22=$4\$2 \cdot 2 = \$4, and so on.

STEP 9

The expected value is calculated just like before: $2116+$2216+$2316+$2416+$2516+$2616\$2 \cdot 1 \cdot \frac{1}{6} + \$2 \cdot 2 \cdot \frac{1}{6} + \$2 \cdot 3 \cdot \frac{1}{6} + \$2 \cdot 4 \cdot \frac{1}{6} + \$2 \cdot 5 \cdot \frac{1}{6} + \$2 \cdot 6 \cdot \frac{1}{6}.

STEP 10

We can simplify this to $2(16+26+36+46+56+66)\$2 (\frac{1}{6} + \frac{2}{6} + \frac{3}{6} + \frac{4}{6} + \frac{5}{6} + \frac{6}{6}).

STEP 11

Adding the fractions, we get 1+2+3+4+5+66=216=72\frac{1+2+3+4+5+6}{6} = \frac{21}{6} = \frac{7}{2}.

STEP 12

So, the expected value is $272=$7\$2 \cdot \frac{7}{2} = \$7.

STEP 13

The expected value of the spinner game is $10\$10, and the expected value of the dice game is $7\$7.

STEP 14

Since $10\$10 is greater than $7\$7, we should take our sister's offer and play the spinner game!

STEP 15

The spinner game has an expected value of $10\$10, while the dice game has an expected value of $7\$7.
Therefore, you should choose to play the spinner game with your sister.

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord