Question

use the fundamental counting principle to find the total number of possible sandwiches: bread: white, wheat, rye meat: turkey, ham, roast beef cheese: american, swiss, muenster what is the probability that your sandwich will have american cheese?______

Explanation:

Step1: Calculate total number of sandwiches

Using the fundamental counting principle, if there are \(n_1\) ways to choose bread (\(n_1 = 3\): white, wheat, rye), \(n_2\) ways to choose meat (\(n_2=3\): turkey, ham, roast - beef) and \(n_3\) ways to choose cheese (\(n_3 = 3\): American, Swiss, muenster). The total number of possible sandwiches \(N=n_1\times n_2\times n_3\).

$$N = 3\times3\times3=27$$

Step2: Calculate number of sandwiches with American cheese

If the cheese is fixed as American (\(n_3'=1\)), and \(n_1 = 3\) (bread options) and \(n_2 = 3\) (meat options). The number of sandwiches with American cheese \(n=n_1\times n_2\times n_3'\)

$$n=3\times3\times1 = 9$$

Step3: Calculate probability

The probability \(P\) of an event is given by the formula \(P=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\). Here, the number of favorable outcomes (sandwiches with American cheese) is \(n = 9\) and the total number of outcomes \(N = 27\)

$$P=\frac{9}{27}=\frac{1}{3}$$

Answer:

\(\frac{1}{3}\)