Ann wants to buy a coffee machine. She knows, that during the holidays, the probability of a discount of more than 10% is 0.87, on weekends, the probability of a discount of more than 10% is 0.38, and on a regular weekday, it is 0.17.
On average there are 15 holidays per year, 100 weekends (holiday weekends are excluded) and 250 weekdays.
Find the probability of a discount of more than 10% for the randomly chosen day.
Find the probability that it was a weekend if it is given that she bought a machine with a discount of more than 10%.
To find the probability of a discount of more than 10% for the randomly chosen day, we need to consider the probabilities of each type of day (holiday, weekend, and regular weekday) and their corresponding discount probabilities.
First, let's find the probability of each type of day occurring:
- Holidays: 15 holidays per year.
- Weekends (excluding holiday weekends): 52 weeks per year - 15 holidays = 37 weekends.
- Regular weekdays: 365 days per year - 15 holidays - 37 weekends = 313 weekdays.
Now, let's calculate the overall probability of each type of day:
- Probability of a holiday: 15 holidays per year / 365 total days per year ≈ 0.0411.
- Probability of a weekend: 37 weekends / 365 total days per year ≈ 0.1014.
- Probability of a regular weekday: 313 weekdays / 365 total days per year ≈ 0.8575.
Next, we multiply the probabilities of each type of day by their corresponding discount probabilities:
- Probability of a discount on a holiday: Probability of a holiday * Probability of a discount > 10% during holidays = 0.0411 * 0.87 = 0.0357.
- Probability of a discount on a weekend: Probability of a weekend * Probability of a discount > 10% during weekends = 0.1014 * 0.38 = 0.0386.
- Probability of a discount on a regular weekday: Probability of a regular weekday * Probability of a discount > 10% on regular weekdays = 0.8575 * 0.17 = 0.1459.
Finally, to find the overall probability of a discount greater than 10% for the randomly chosen day, we sum up the probabilities of each type of day:
Overall probability = Probability of a discount on a holiday + Probability of a discount on a weekend + Probability of a discount on a regular weekday
= 0.0357 + 0.0386 + 0.1459
= 0.2202.
Therefore, the probability of a discount of more than 10% for the randomly chosen day is approximately 0.2202, or 22.02%.
To find the probability that it was a weekend if it is given that she bought a machine with a discount of more than 10%, we can use Bayes' theorem.
Let's define the events:
A: Buying a machine with a discount of more than 10%.
B: It was a weekend.
We need to find P(B|A), the probability that it was a weekend given that she bought a machine with a discount of more than 10%.
We know that P(A) (the probability of buying a machine with a discount of more than 10%) is given by the overall probability calculated previously, which is 0.2202.
Now, we need to find P(A|B), the probability of buying a machine with a discount of more than 10% given that it was a weekend.
Using Bayes' theorem:
P(A|B) = [P(B|A) * P(A)] / P(B).
We have already calculated P(A), and we can calculate P(B) as the probability of a weekend, which is 0.1014.
Now, we need to calculate P(B|A), the probability that it was a weekend given that she bought a machine with a discount of more than 10%.
P(B|A) = [P(A|B) * P(B)] / P(A).
Assuming that the probability of a discount on a weekend is independent of the type of day, we can write P(A|B) = P(A) = 0.2202.
Substituting the known values:
P(B|A) = (0.2202 * 0.1014) / 0.2202
= 0.0223.
Therefore, the probability that it was a weekend given that she bought a machine with a discount of more than 10% is approximately 0.0223, or 2.23%.