A pizza parlor has 12 different toppings available for its pizzas, and 2 of these toppings are pepperoni and anchovies. If a customer picks 2 toppings at random, find the probability that neither topping is anchovies.
Round your answer to four decimal places.
first pick
p anchovies = 1/12
p not anchovies = 11/12
second pick
you now only have 11 toppings one of which is anchovies
p anchovies = 1/11
p not anchovies = 10/11
11*10/(11*12) = 10/12 = 5/6
p neither anchovies =
To find the probability that neither topping is anchovies, we need to calculate the number of favorable outcomes (where neither topping is anchovies) and divide it by the total number of possible outcomes.
Step 1: Calculate the total number of possible outcomes.
Since the customer picks 2 toppings out of 12 available, we will use the combination formula. The total number of possible outcomes is given by:
nCr = n! / (r!(n-r)!)
Here, n = 12 (total number of toppings) and r = 2 (number of chosen toppings).
12C2 = 12! / (2!(12-2)!)
= 12! / (2! * 10!)
= (12 * 11 * 10!) / (2! * 10!)
= (12 * 11) / 2
= 132 / 2
= 66
Therefore, the total number of possible outcomes is 66.
Step 2: Calculate the number of favorable outcomes.
Since we want to exclude anchovies, there are 11 toppings remaining out of which the customer can choose 2. We will again use the combination formula:
11C2 = 11! / (2!(11-2)!)
= 11! / (2! * 9!)
= (11 * 10 * 9!) / (2! * 9!)
= (11 * 10) / 2
= 110 / 2
= 55
Therefore, the number of favorable outcomes is 55.
Step 3: Calculate the probability.
The probability is given by:
Probability = favorable outcomes / total outcomes
Probability = 55 / 66
Now, rounding our answer to four decimal places:
Probability ≈ 0.8333
To find the probability that neither topping is anchovies, we need to determine the total number of possible combinations of 2 toppings and the number of combinations that do not include anchovies.
First, let's calculate the total number of possible combinations of 2 toppings. This can be done using the formula for combinations, which is given by:
nCr = n! / (r!(n - r)!)
In this case, we have 12 toppings and we want to choose 2, so n = 12 and r = 2. Plugging these values into the formula, we get:
12C2 = 12! / (2!(12 - 2)!)
= 12! / (2! * 10!)
= (12 * 11 * 10!) / (2! * 10!)
= 12 * 11 / 2
= 66
There are 66 possible combinations of 2 toppings.
Next, let's determine the number of combinations that do not include anchovies. Since we have 2 toppings and only 1 of them is anchovies, we need to choose the remaining topping from the remaining 11 options (excluding anchovies). This can be calculated using the combination formula:
11C2 = 11! / (2!(11 - 2)!)
= 11! / (2! * 9!)
= (11 * 10 * 9!) / (2! * 9!)
= 11 * 10 / 2
= 55
There are 55 combinations of 2 toppings that do not include anchovies.
Finally, we can calculate the probability by dividing the number of combinations that do not include anchovies by the total number of possible combinations:
P(neither topping is anchovies) = 55/66
Rounding the answer to four decimal places, we get:
P(neither topping is anchovies) = 0.8333