Gerald found a $24.99 backpack that he’d like to purchase from the sporting goods store. The current promotion involves scratching a card to reveal a discount that can be 5% off, 15% off or 25% off. Each of the three discount options is equally likely. After the discount is taken, a 6.5% tax is added. What is the probability that Gerald will pay less than $25 to purchase this backpack? Express your answer as a common fraction.
clearly, two of the three possible discounts will give enough discount so that the final price with tax will be less than $25.
So, probability is 2/3
your momy
To determine the probability that Gerald will pay less than $25 to purchase this backpack, we need to consider the possible discounts and the final price after tax.
Let's list the possible discounts:
- 5% off
- 15% off
- 25% off
The probability of each discount option is equally likely, meaning each has a probability of 1/3.
Now, let's calculate the price after the discount is applied:
- For a 5% discount, the price after discount is (100% - 5%) * $24.99 = 95% * $24.99.
- For a 15% discount, the price after discount is (100% - 15%) * $24.99 = 85% * $24.99.
- For a 25% discount, the price after discount is (100% - 25%) * $24.99 = 75% * $24.99.
Next, let's calculate the final price after the 6.5% tax is added:
- For a 5% discount, the final price is (1 + 6.5%) * (95% * $24.99).
- For a 15% discount, the final price is (1 + 6.5%) * (85% * $24.99).
- For a 25% discount, the final price is (1 + 6.5%) * (75% * $24.99).
Finally, we need to determine the probability of each final price being less than $25 and sum them up.
Calculating the probabilities:
- For a 5% discount: The probability is 1/3 * (probability of final price less than $25 after tax).
- For a 15% discount: The probability is 1/3 * (probability of final price less than $25 after tax).
- For a 25% discount: The probability is 1/3 * (probability of final price less than $25 after tax).
To calculate the probability of each final price being less than $25, divide the maximum price less than $25 by the final price after tax and round it to the nearest whole number:
- For a 5% discount:
- Maximum price less than $25 = $25 - ($25 * 6.5%).
- Probability of final price less than $25 after tax = Maximum price less than $25 / (1 + 6.5%) * (95% * $24.99).
- For a 15% discount:
- Maximum price less than $25 = $25 - ($25 * 6.5%).
- Probability of final price less than $25 after tax = Maximum price less than $25 / (1 + 6.5%) * (85% * $24.99).
- For a 25% discount:
- Maximum price less than $25 = $25 - ($25 * 6.5%).
- Probability of final price less than $25 after tax = Maximum price less than $25 / (1 + 6.5%) * (75% * $24.99).
Adding up the probabilities for each discount option will give us the final probability that Gerald will pay less than $25 to purchase the backpack.