An ice-cream parlor sells sundaes with 3 different types of ice-cream and 4 different toppings. They have 8 flavors of ice-cream and 10 toppings for their customers to chose from.
What is the probability that a customer orders a sundae including the toppings of whipped cream, caramel syrup, chocolate syrup, and cookies crumbles?
To calculate the probability that a customer orders a sundae with whipped cream, caramel syrup, chocolate syrup, and cookie crumbles, we need to determine the number of favorable outcomes (sundaes with all four toppings) and the total number of possible outcomes.
First, we need to determine the number of favorable outcomes:
Number of ways to choose the ice cream flavor: Since there are 8 flavors of ice cream available and the customer can choose any 3, the number of ways to choose the ice cream flavor would be 8 choose 3, denoted as C(8, 3) or 8C3. This can be calculated as:
C(8, 3) = 8! / (3! * (8-3)!) = 8! / (3! * 5!) = (8 * 7 * 6) / (3 * 2) = 56
Number of ways to choose the toppings: Since there are 10 toppings available and the customer needs to choose 4, the number of ways to choose the toppings would be 10 choose 4, denoted as C(10, 4) or 10C4. This can be calculated as:
C(10, 4) = 10! / (4! * (10-4)!) = 10! / (4! * 6!) = (10 * 9 * 8 * 7) / (4 * 3 * 2 * 1) = 210
Now, we can calculate the total number of possible outcomes:
Number of ways to choose the ice cream flavor: Same as before, C(8, 3) = 56
Number of ways to choose the toppings: Same as before, C(10, 4) = 210
Finally, we can calculate the probability:
Probability = (Number of favorable outcomes) / (Total number of possible outcomes) = (56 * 1) / (56 * 210) = 1 / 210 ≈ 0.004762
Therefore, the probability that a customer orders a sundae including whipped cream, caramel syrup, chocolate syrup, and cookie crumbles is approximately 0.004762 or about 0.48%.
To find the probability of a customer ordering a sundae with specific toppings, we need to calculate the ratio of the number of favorable outcomes (sundaes with the desired toppings) to the total number of possible outcomes (all possible sundaes).
First, let's calculate the number of favorable outcomes. Since the customer wants whipped cream, caramel syrup, chocolate syrup, and cookie crumbles, we need to choose one flavor of ice cream from the 8 available flavors and one of each topping from the 10 available toppings.
The number of favorable outcomes = 8 (flavors of ice cream) * 1 (whipped cream) * 1 (caramel syrup) * 1 (chocolate syrup) * 1 (cookie crumbles) = 8
Next, let's calculate the total number of possible outcomes. The customer can choose any combination of 3 types of ice cream out of the 8 available flavors, and any combination of 4 toppings out of the 10 available options.
The total number of possible outcomes = C(8, 3) (combinations of 3 ice cream flavors) * C(10, 4) (combinations of 4 toppings)
Using the combination formula, C(n, r) = n! / (r! * (n-r)!), where n is the total number options and r is the number to choose, we can calculate the total number of possible outcomes:
C(8, 3) = 8! / (3! * (8-3)!) = 8! / (3! * 5!) = (8 * 7 * 6) / (3 * 2 * 1) = 56
C(10, 4) = 10! / (4! * (10-4)!) = 10! / (4! * 6!) = (10 * 9 * 8 * 7) / (4 * 3 * 2 * 1) = 210
The total number of possible outcomes = C(8, 3) * C(10, 4) = 56 * 210 = 11,760
Finally, we can calculate the probability by dividing the number of favorable outcomes by the total number of possible outcomes:
Probability = Number of favorable outcomes / Total number of possible outcomes
= 8 / 11,760
≈ 0.00068 (rounded to 5 decimal places)
Therefore, the probability that a customer orders a sundae including whipped cream, caramel syrup, chocolate syrup, and cookie crumbles is approximately 0.00068.