Question

A bowl contains 16 chips, of which 6 are red, 7 are white, and 3 are blue. If four chips are talsen at random and without replacement, find the probability that: (a) each of the 4 chips is red; (b) none of the 4 chips is red; (c) there is at least 1 chip of each color.

Short answer

The solutions are: (a) The probability that each of the 4 chips is red is given by \( P(A) = \frac{_{6}C_{4}}{_{16}C_{4}} \); (b) The probability that none of the 4 chips is red is given by \( P(B) = \frac{_{10}C_{4}}{_{16}C_{4}} \); and (c) The probability that there is at least 1 chip of each color is given by \( P(C) = \frac{_{6}C_{1} \times _{7}C_{1} \times _{3}C_{1} \times _{16}C_{4}}{_{16}C_{4}} \)

Step-by-step solution

Possible outcomes

Compute the total number of possible outcomes when drawing four chips without replacement from sixteen chips. This is calculated as the combination of 16 items taken 4 at a time, denoted by \( _{16}C_{4} = \frac{16!}{(16-4)!4!} \)

Calculating condition (a)

To find the probability of drawing four red chips, calculate the total number of combinations of obtaining 4 red chips from 6, which is denoted as \( _6C_4 = \frac{6!}{(6-4)!4!} \). The probability of this event is the ratio of the favorable outcomes to the total outcomes, hence \( P(A) = \frac{_{6}C_{4}}{_{16}C_{4}} \)

Calculating condition (b)

To compute the probability of drawing no red chip, calculate the total combinations of drawing four chips from the non-red chips, which is 10 in number. This is denoted as \( _{10}C_{4} = \frac{10!}{(10-4)!4!} \). The probability of this event is then \( P(B) = \frac{_{10}C_{4}}{_{16}C_{4}} \)

Calculating condition (c)

Finding the probability of drawing at least one chip of each color involves a combination of events. Consider cases where you draw one chip of each color and the fourth chip being any color. This can be calculated as \( P(C) = \frac{_{6}C_{1} \times _{7}C_{1} \times _{3}C_{1} \times _{16}C_{4}}{_{16}C_{4}} \)

Key concepts

Combinatorial Probability

Combinatorial probability is a branch of mathematics that deals with the likelihood of the occurrence of a particular combination of events. It involves counting the number of ways certain events can happen, particularly in situations where events are dependent on one another, such as when drawing objects from a set without replacement.

For example, in the context of the provided exercise where chips are drawn from a bowl, combinatorial probability allows us to calculate the chances of drawing chips of specific colors in a certain order. This is typically done by computing combinations, which are selections in which the order does not matter. The formula for combinations is denoted as \( _nC_k = \frac{n!}{(n-k)!k!} \) where \( n \) is the total number of items, and \( k \) is the number of items to choose. In this formula, \( ! \) represents the factorial operation, multiplying the number by all positive integers less than itself.

For instance, when computing the probability of drawing four red chips from six without replacement, as in part (a) of the exercise, we would calculate the combinations of 6 taken 4 at a time: \( _{6}C_{4} \) and compare it to the total possible combinations of 16 chips taken 4 at a time: \( _{16}C_{4} \). This approach to probability is extremely useful in a variety of fields, from card games to data analysis in scientific research.

Probability Theory

Probability theory is the mathematical framework that allows us to analyze random events and predict the likelihood of various outcomes. It encompasses techniques and principles that help quantify uncertainty and form the foundation for statistical inference.

The very core of probability theory starts with understanding events, outcomes, and the probabilities associated with them. An 'event' is a result that we can get from a random experiment, while an 'outcome' is a specific result from the set of all possible results, known as the sample space. The probability of an event occurring is a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.

When dealing with exercises like the chip-drawing problem, probability theory guides us on how to use formulas and reason to calculate these probabilities. For example, drawing four chips without replacement is a dependent event; each draw affects the next since the chips are not returned to the bowl. This level of complexity adds richness to the field of probability and challenges students to think carefully about the relationships between events as they calculate probabilities.

Mathematical Statistics

Mathematical statistics is a branch of mathematics that focuses on the collection, analysis, interpretation, and presentation of data. It relies heavily on probability theory as it deals with making inferences from sample data to entire populations.

In the context of the problem at hand, once we have found the probabilities for different events, we may use these findings to make informed statements about larger patterns or trends. For instance, if we were to repeatedly draw sets of four chips from this bowl, we could start to predict the distribution of color outcomes over time.

Mathematical statistics involves both descriptive statistics, which summarize the data at hand, such as through the use of means and variances, and inferential statistics, which draw predictions and conclusions about a larger population. It's crucial in fields that depend on data-driven decision making, such as finance, healthcare, and public policy. By understanding the statistical significance and reliability of our results, we can make more accurate and responsible interpretations of data — whether we're considering a simple game involving chips or complex real-world scenarios.