To promote a new brand of shoes, a shoe store will run a promotion using a jar containing 3 red balls marked '10% off,' 2 white balls marked '30% off,' and 1 green ball marked '60%... To promote a new brand of shoes, a shoe store will run a promotion using a jar containing 3 red balls marked '10% off,' 2 white balls marked '30% off,' and 1 green ball marked '60% off.' Each customer will randomly select 1 ball from the jar to determine the discount that the customer will receive on any single pair of the new brand of shoes. Given that the new brand of shoes regularly costs $60 per pair, what is the average discount amount, in dollars, that the store can expect to give each customer due to this promotion?
Understand the Problem
The question is asking to determine the average discount amount in dollars that a shoe store can expect to give to each customer based on a promotional system of selecting balls with different discount values. This involves calculating the expected value of the discounts based on the probabilities of drawing each type of ball.
Answer
$15$
Answer for screen readers
The average discount amount that the store can expect to give each customer due to this promotion is $15.
Steps to Solve
- Identify the Discounts and Their Values
The discounts offered by the balls are:
- 3 red balls: $10%$ off
- 2 white balls: $30%$ off
- 1 green ball: $60%$ off
- Determine Total Number of Balls
Add the total number of balls:
$$
3 \text{ (red)} + 2 \text{ (white)} + 1 \text{ (green)} = 6 \text{ balls}
$$
- Calculate the Probability of Each Discount
The probability of drawing each color of ball:
- Red (10%):
$$ P(\text{Red}) = \frac{3}{6} = \frac{1}{2} $$ - White (30%):
$$ P(\text{White}) = \frac{2}{6} = \frac{1}{3} $$ - Green (60%):
$$ P(\text{Green}) = \frac{1}{6} $$
- Calculate the Monetary Value of Each Discount
The monetary value of each discount for a $60$ pair of shoes:
- Red (10% off):
$$ \text{Value}_{\text{Red}} = 0.10 \times 60 = 6 $$ - White (30% off):
$$ \text{Value}_{\text{White}} = 0.30 \times 60 = 18 $$ - Green (60% off):
$$ \text{Value}_{\text{Green}} = 0.60 \times 60 = 36 $$
- Calculate Expected Value of the Discounts
Use the probabilities and values to find the expected discount:
$$
E = P(\text{Red}) \cdot \text{Value}{\text{Red}} + P(\text{White}) \cdot \text{Value}{\text{White}} + P(\text{Green}) \cdot \text{Value}_{\text{Green}}
$$
Filling in the values:
$$
E = \left(\frac{1}{2} \cdot 6\right) + \left(\frac{1}{3} \cdot 18\right) + \left(\frac{1}{6} \cdot 36\right)
$$
Calculating each term:
$$
E = 3 + 6 + 6 = 15
$$
The average discount amount that the store can expect to give each customer due to this promotion is $15.
More Information
The expected value is a statistical measure that computes the weighted average of all possible values, considering their probabilities. In this case, it's used to determine how much discount the store anticipates providing on average per customer.
Tips
- Forgetting to adjust the discount percentages to their dollar values based on the price of the shoes. Always convert the percentage discount into actual dollar amounts.
- Overlooking the total number of balls and miscalculating the probabilities. Ensure that the denominator is the total of all balls.
AI-generated content may contain errors. Please verify critical information
