Question

There are 5 red chips and 3 blue chips in a bowl. The red chips are numbered \(1,2,3,4,5\), respectively, and the blue chips are numbered \(1,2,3\), respectively. If 2 chips are to be drawn at random and without replacement, find the probability that these chips have either the same number or the same color.

Short answer

The probability that the drawn chips have either the same number or the same color is \( \frac{2}{7} \).

Step-by-step solution

Identify the Total Possible Outcome

In total, there are 8 chips. If 2 chips are to be drawn, there are \( \binom{8}{2} = 28 \) ways.

Identify Favorable Outcomes

The favorable events are that the chips drawn have either the same number or the same color. We calculate these two possibilities separately and then combine, as they don't occur all at the same time. (i) Same number: Each number from 1 to 3 has 2 chips with the same number, so there are \( \binom{2}{2}*3 = 3 \) ways. For the numbers 4 and 5, since they are all red, there are \( \binom{2}{2}*2 = 2 \) ways.(ii) Same color: This excludes the favorable outcomes we have already considered in (i). For red, in addition to the two with the number 4 and 5, there are \( \binom{3}{2} = 3 \) ways. For blue, there are no more left.So in total, there are \( 3+2+3 = 8 \) ways.

Calculate the Probability

Now we can calculate the probability. The number of favorable outcomes is 8, and the number of total possible outcomes is 28. So the probability is \( \frac{8}{28} = \frac{2}{7} \).

Key concepts

Combinatorics

Combinatorics is a branch of mathematics that deals with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is related to many other areas of mathematics, such as algebra, probability, and geometry, and has applications in fields as diverse as computer science and statistical physics.

At its core, combinatorics is about finding the number of ways certain arrangements or selections can be made. The basic concepts include permutations, which are arrangements of objects where the order matters, and combinations, which refer to selections where order does not matter.

In the context of our exercise, combinations are used, denoted by the 'choose' function (i.e., \( \binom{n}{r} \)), which calculates the number of ways to choose \(r\) objects from a set of \(n\) without regard to the order. This is crucial for solving problems that deal with the selection of objects, like drawing chips from a bowl, where we're interested in the selection and not the sequence in which they're drawn.

Random Sampling Without Replacement

Random sampling without replacement is a statistical method used to randomly select a subset of individuals from a larger population where each individual can be selected at most once. This ensures that each individual has an equal chance of being selected in each draw, but once selected, it is removed from the pool and cannot be picked again.

In our exercise, this concept is illustrated by drawing chips from a bowl. When a chip is drawn, it is not returned to the bowl; thus, it cannot be selected again. This changes the probability of the subsequent draws because the total number of chips decreases, affecting the probability calculations. Random sampling without replacement is common in situations where it is important not to count the same subject twice, which is often required in probability games, lotteries, and research studies where each subject must only contribute once.

Probability Calculations

Probability calculations are a fundamental part of combinatorics and statistics. They involve determining the likelihood of a particular event occurring. The calculation can be as simple as dividing the number of ways the desired event can occur by the total number of possible events (provided all events are equally likely to occur).

In calculating probabilities, we often encounter two types of events: 'favorable' and 'possible'. Favorable events are the outcomes we are interested in, while possible events represent all the outcomes that could possibly occur. The probability of an event is given by the formula: \(\text{Probability} = \frac{{\text{Number of favorable outcomes}}}{{\text{Number of total possible outcomes}}}\). For our chips drawing exercise, we computed the probability of drawing two chips with either the same number or color. By identifying and counting these favorable conditions, and considering all possible outcomes, we obtained the probability which helps in the understanding of how likely an event is to happen under the given conditions.