Manuela and Stephen survey 250 people at a sporting event and ask if they prefer hamburgers or hot dogs, and if they prefer regular or diet soda.

Regular Soda Diet Soda Total

Hamburgers 90 40 130

Hot Dogs 70 50 120

Total 160 90 250

a. What is the probability that someone who prefers diet soda will also prefer hamburgers?

b. What is the probability that someone who prefers hot dogs will also prefer regular soda?

c. Are preferring diet soda and preferring hamburgers independent events? Explain.

d. Are preferring regular soda and preferring hot dogs independent events? Explain.

a. 40/250

b. 70/250

c. Preferring diet soda and preferring hamburgers are not independent events

d.  Preferring regular soda and preferring hot dogs are not independent events

Step-by-step explanation:

We are given:

                      Regular Soda    Diet Soda    Total

Hamburgers             90                 40            130

Hot Dogs                   70                 50           120

Total                          160                90          250

a. The probability that someone who prefers diet soda will also prefer hamburgers can be calculated as:

No. of people who prefer hamburgers and diet soda / Total no. of people

40/250

b. The probability that someone who prefers hot dogs will also prefer regular soda can be calculated as:

No. of people who prefer hot dogs and regular soda/Total no. of people

70/250

c. For two events to be independent, they must fulfill the condition:

P(A and B) = P(A) * P(B)

P(Diet Soda) = No. of people who prefer diet soda/total no. of people

                     = 90/250

P(Hamburgers) = No. of people who prefer hamburgers/total no. of people

                     = 130/250

P(Diet Soda and Hamburgers) = 40/250

Now, we need to check:

P(Diet Soda and Hamburgers) = P(Diet Soda) * P(Hamburgers)

40/250 = 90/250 * 130/250

40/250 ≠ 11700/62500

So, we can conclude that these two events are not independent events.

d. P(Regular Soda) = No. of people who prefer regular soda/total no. of people

                     = 160/250

   P(Hot dogs) = No. of people who prefer hot dogs/total no. of people

                     = 120/250

P(Regular Soda and Hot Dogs) = 70/250

Now, we need to check if:

P(Regular Soda and Hot Dogs) = P(Regular Soda) * P(Hot dogs)

70/250 ≠ 160/250 * 120/250

So, these two events are not independent.