Questions: Due to a manufacturing error, three cans of regular soda were accidentally filled with diet soda and placed alongside cans of regular soda in an 18-pack. Suppose that two cans are randomly selected from the 18-pack. Complete parts (a) through (c). (a) Determine the probability that both contain diet soda. P(both diet)=0.0196 (Round to four decimal places as needed.) (b) Determine the probability that both contain regular soda. P(both regular)= (Round to four decimal places as needed.)

Solution Steps

To find the probability that both selected cans contain diet soda, we use the hypergeometric distribution. The parameters are as follows:

• Total number of cans, \( N = 18 \)
• Number of diet soda cans, \( K = 3 \)
• Number of cans drawn, \( n = 2 \)
• Number of diet soda cans in the sample, \( k = 2 \)

The calculated probability is: \[ P(\text{both diet}) = 0.0196 \]

Next, we calculate the probability that both selected cans contain regular soda. The parameters for this calculation are:

• Total number of cans, \( N = 18 \)
• Number of regular soda cans, \( K = 15 \) (since \( 18 - 3 = 15 \))
• Number of cans drawn, \( n = 2 \)
• Number of regular soda cans in the sample, \( k = 2 \)

The calculated probability is: \[ P(\text{both regular}) = 0.6863 \]

Final Answer