During a dinner party, Magda plans on opening six bottles of...

During a dinner party, Magda plans on opening six bottles of wine. Her supply includes 8 French, 10 Australian and 12 Italian wines. She sends her sister Mara to choose the bottles. Mara has no knowledge of the wine types and picks the bottles at random. a) What is the probability that two of each type get selected? b) What is the probability that all served bottles are of the same type? c) What is the probability of serving only Italian and French wines?

During a dinner party, Magda plans on opening six bottles of wine. Her supply includes 8 French, 10 Australian and 12 Italian wines. She sends her sister Mara to choose the bottles. Mara has no knowledge of the wine types and picks the bottles at random.
a) What is the probability that two of each type get selected?
b) What is the probability that all served bottles are of the same type?
c) What is the probability of serving only Italian and French wines?

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Key Concepts

Sampling Without Replacement

This concept refers to the process of drawing items from a set without putting them back after each selection. In probability problems like these, it is important because the outcome of one draw affects the subsequent draws, thereby altering the probabilities as the sample space changes.

Combination Counting

Combination counting involves using binomial coefficients to calculate the number of ways to select k items from a set of n items without considering the order. This key concept is essential for determining both the total number of outcomes and the number of favorable outcomes in selection problems.

Favorable Versus Total Outcomes

A fundamental idea in probability is to determine the ratio of favorable outcomes to the total number of possible outcomes. This requires careful counting of the specific events that meet the criteria of the problem and using combinatorial methods to compute both quantities.

Hypergeometric Distribution

The hypergeometric distribution applies to scenarios where a sample is drawn from a finite population without replacement. It is used to calculate the probability of a specific combination of successes and failures when the population contains items of different types, making it directly relevant to problems involving random selection from distinct subgroups.

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