Question

a cooler contains ten bottles of sports drinks: three lemon - lime flavored, two orange flavored, and five fruit - punch flavored. three times, you randomly grab a bottle, return the bottle to the cooler and then mix up the bottles. the first time, you get lemon - lime. the second time and third times you get fruit - punch. what is the probability that this event happens? enter your answer as a simplified fraction.

Explanation:

Step1: Find Probability of Lemon - Lime

There are 3 lemon - lime bottles out of 10 total. So the probability of getting lemon - lime, \(P(L)=\frac{3}{10}\).

Step2: Find Probability of Fruit - Punch

There are 5 fruit - punch bottles out of 10 total. So the probability of getting fruit - punch, \(P(F)=\frac{5}{10}=\frac{1}{2}\).

Step3: Calculate Combined Probability

Since the events are independent (we return the bottle each time), we multiply the probabilities of each event. The event is lemon - lime first, then fruit - punch, then fruit - punch. So the probability \(P = P(L)\times P(F)\times P(F)\).
Substitute the values: \(P=\frac{3}{10}\times\frac{5}{10}\times\frac{5}{10}\).
First, multiply \(\frac{3}{10}\times\frac{5}{10}=\frac{15}{100}\), then multiply by \(\frac{5}{10}\): \(\frac{15}{100}\times\frac{5}{10}=\frac{75}{1000}\). Simplify the fraction by dividing numerator and denominator by 25: \(\frac{75\div25}{1000\div25}=\frac{3}{40}\).

Answer:

\(\frac{3}{40}\)