Problem 36 - A History of Mathematics: An Introduction

Question

There are 12 balls in an urn, 4 of which are white and 8 black. Three blindfolded players, \(A, B, C\) draw a ball in. turn, first \(A\), then \(B\), then \(C\). The winner is the one who first draws a white ball. Assuming that each (black) ball is replaced after being drawn, find the ratio of the chances of the three players.

Step-by-Step Solution

Answer

Answer: The ratio of chances for players A, B, and C is 9:6:4.

Step 1: Calculate the Probability of Player A Winning

Player A wins if they draw a white ball on the first draw. The probability that Player A draws a white ball on their first turn is the ratio of white balls to total balls. There are 4 white balls and 12 total balls, so the probability that player \(A\) wins is \(\frac{4}{12}=\frac{1}{3}\).

Step 2: Calculate the Probability of Player B Winning

Player B wins if Player A first draws a black ball, and then Player B draws a white ball. The probability that Player A draws a black ball on their first turn is \(\frac{8}{12}=\frac{2}{3}\). Then, the probability that Player B draws a white ball after Player A has drawn a black ball is \(\frac{4}{12}=\frac{1}{3}\). Since these events are independent, we multiply the probabilities together to get the probability of Player B winning: \(\frac{2}{3} \cdot \frac{1}{3}=\frac{2}{9}\).

Step 3: Calculate the Probability of Player C Winning

Player C wins if Player A firsts draws a black ball, followed by Player B drawing a black ball, and then Player C drawing a white ball. We already know the probability of both players A and B drawing black balls: \(\frac{8}{12}=\frac{2}{3}\). Therefore, the probability of both events happening one after another is \((\frac{2}{3})^2 = \frac{4}{9}\). Then, the probability of Player C drawing a white ball after both Players A and B drew black balls is \(\frac{4}{12}=\frac{1}{3}\). Since these events are independent, we multiply the probabilities together to get the probability of Player C winning: \(\frac{4}{9} \cdot \frac{1}{3}=\frac{4}{27}\).

Step 4: Determine the Ratio of Chances

Now we know the probability of each player winning: Player A has a \(\frac{1}{3}\) chance, Player B has a \(\frac{2}{9}\) chance, and Player C has a \(\frac{4}{27}\) chance. To find the ratio of their chances, we express these probabilities in a form where they share a common denominator. The common denominator for these probabilities is 27. We find the ratio of the chances to be: \(\frac{(1/3) \cdot 27}{27} : \frac{(2/9) \cdot 27}{27} : \frac{(4/27) \cdot 27}{27}\) = \(9:6:4\). That is the ratio of chances for players \(A\), \(B\), and \(C\) respectively.

Key Concepts

Probability Theory

Probability theory is a mathematical framework for quantifying uncertainty and making predictions about the likelihood of various outcomes. In our exercise involving the drawing of balls from an urn, we use probability theory to determine the chances of each player winning the game.

The fundamental principle here is the calculation of the probability of an event, which is the number of favorable outcomes divided by the total number of possible outcomes. For instance, when Player A draws a ball, the probability that the ball is white is calculated by dividing the number of white balls (4) by the total number of balls (12), giving \(\frac{4}{12} = \frac{1}{3}\).

In these calculations, understanding the basic principles, such as the concept of favorable versus possible outcomes, is crucial for accurately assessing chances in any probabilistic scenario.

Independent Events

Independent events in probability are those whose outcome does not affect the outcome of another event. This concept is essential when calculating the probability of a sequence of events, like in our urn problem.

For Player B to win, Player A must first draw a black ball, and then Player B must draw a white ball. These two events are independent because the drawing of one ball does not influence the drawing of the next, especially since each ball is replaced after being drawn. This allows us to multiply the probabilities of the two independent events to find the overall probability of Player B winning: \(\frac{2}{3} \times \frac{1}{3} = \frac{2}{9}\).

The concept of independent events is widely applicable in games of chance, as it allows us to methodically analyze sequences of actions without worrying about previous outcomes affecting future ones.

Mathematical Statistics

Mathematical statistics involves collecting, analyzing, interpreting, presenting, and organizing data. In the context of probability games like our exercise, it includes using probabilities to make predictions about an event’s outcome.

In the case of our blindfolded players, mathematical statistics would not only help us determine the probabilities of each player drawing a white ball but also inform us about the expected distribution of wins over a large number of games. If a large group of people played this game repeatedly under the same conditions, we'd expect Player A to win roughly \(\frac{1}{3}\) of the time, Player B about \(\frac{2}{9}\), and Player C approximately \(\frac{4}{27}\) of the games based on the probabilities we've calculated.

Combinatorics

Combinatorics is a branch of mathematics dealing with counting, combination, and permutation of sets of elements, and it is fundamental in calculating probabilities in games. In our urn example, combinatorics isn't explicitly needed because we're dealing with directly calculable probabilities based on single draws.

However, if we were to consider a more complex scenario where, for example, the players could draw multiple balls without replacement, combinatorial analysis would help us count the possible outcomes and determine probabilities. Through combinatorial principles, we can count complex events' possibilities where order and selection without replacement become significant factors.