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Chapter 3: Problem 47

An urn contains 5 white and 10 black balls. A fair die is rolled and that number of balls is randomly chosen from the urn. What is the probability that all of the balls selected are white? What is the conditional probability that the die landed on 3 if all the balls selected are white?

Short Answer

Expert verified
The probability of selecting all white balls is \[ P(\text{All white}) = \frac{C(5, n)}{C(15, n)} \times \frac{1}{6}, \] summed for all roll values \(n = 1, \dots, 6\). The conditional probability that the die landed on 3 if all the balls selected are white is \[ P(\text{Roll 3} | \text{All white}) = \frac{P(\text{All white} | \text{Roll }= 3) \times P(\text{Roll }= 3)}{P(\text{All white})}. \]

Step by step solution

01

Calculate the probability of selecting all white balls for each roll value

To do this, we'll calculate the probability of selecting only white balls for each possible roll value, i.e., n = 1, 2, 3, 4, 5, and 6. Since there are a total of \(15\) balls in the urn, with \(5\) being white and \(10\) being black, for each possible roll value, the probability of selecting all white balls can be calculated as a ratio of the number of ways of choosing only white balls to the number of ways of choosing any \(n\) balls. We can use the combination formula to calculate the number of ways of selecting balls: \[ C(n, k) = \frac{n!}{k!(n - k)!}, \] where \(n\) is the total number of items, \(k\) is the number of items to be selected, and ! denotes the factorial of a number.
02

Calculate the weighted probability

Since there are 6 possible roll values and each has equal probability (\(\frac{1}{6}\)), we need to find the weighted probability of selecting all white balls based on each roll value. The probability for each roll value can be calculated as: \[ P(\text{All white} | \text{Roll }= n) = \frac{C(5, n)}{C(15, n)} \times \frac{1}{6}. \] Summing the probabilities for all 6 roll values will give us the total probability of selecting all white balls.
03

Calculate the conditional probability

Now that we have the probabilities for each roll value, let's calculate the conditional probability that the die landed on 3 if all the balls selected are white. We can use the formula for conditional probability: \[ P(\text{Roll 3} | \text{All white}) = \frac{P(\text{All white} | \text{Roll }= 3) \times P(\text{Roll }= 3)}{P(\text{All white})}. \] We already have each of these probabilities except for \(P(\text{All white})\). We can calculate this by summing the probabilities for each roll value as calculated in step 2. Once we have all the probabilities, we plug them into the formula to get the conditional probability.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Conditional Probability
Understanding conditional probability is essential for calculating how likely an event is to occur given that another event has already happened. In our exercise, we explore the probability of rolling a specific number on a die (in this case, 3) given that all the chosen balls are white.

To do this, we use the conditional probability formula, where we need the probability of both events happening together and the probability of the condition alone. For this scenario, we already know all selected balls are white, and we need to find the chance that this happened when a 3 was rolled. This concept allows us to refine our probability estimates based on new information or conditions.
Combinatorics
In the realm of combinatorics, we deal with counting arrangements and combinations where order doesn't matter. With our urn filled with black and white balls, we use combinatorial mathematics to calculate the different ways we can draw a certain number of balls.

Combining Balls

If we want to select balls without concern for the order, we use the combinations formula. For example, whether we select white ball A before white ball B, or vice versa, it's the same combination. Combinatorics is widely used in probability calculations to determine the total possible outcomes, which is a critical step in our problem-solving process.
Factorials in Probability
Mathematics often uses factorials to simplify counting in probability questions. A factorial, denoted by an exclamation point (!), is the product of all positive integers up to a given number. For instance, the factorial of 5, written as \(5!\), is \(5 \times 4 \times 3 \times 2 \times 1 = 120\).

In probability, factorials help us to calculate combinations and permutations. When choosing from a set, the factorial indicates the total number of ways to arrange or select items, as seen in the combinations formula used in our exercise. Grasping the concept of factorials is critical for dissecting the complexities of probability and making sense of potential outcomes.
Weighted Probability
Sometimes, different outcomes have different chances of occurring. This is where weighted probability comes into play, assigning weightings to various probabilities based on certain conditions. In our exercise, each die roll from 1 to 6 has an equal chance of happening, thus each possibility has an equal weighting when determining the overall probability of selecting all white balls.

The process involves multiplying the probability of each individual event by its relative weight (in this case, the probability of rolling a specific number on the die) and summing the results to obtain a comprehensive weighted probability. This allows us to account for different scenarios in a single probability calculation.

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Most popular questions from this chapter

There are 15 tennis balls in a box, of which 9 have not previously been used. Three of the balls are randomly chosen, played with, and then returned to the box. Later, another 3 balls are randomly chosen from the box. Find the probability that none of these balls has ever been used.

A certain organism possesses a pair of each of 5 different genes (which we will designate by the first 5 letters of the English alphabet). Each gene appears in 2 forms (which we designate by lowercase and capital letters). The capital letter will be assumed to be the dominant gene, in the sense that if an organism possesses the gene pair \(x X,\) then it will outwardly have the appearance of the \(X\) gene. For instance, if \(X\) stands for brown eyes and \(x\) for blue eyes, then an individual having either gene pair \(X X\) or \(x X\) will have brown eyes, whereas one having gene pair \(x x\) will have blue eyes. The characteristic appearance of an organism is called its phenotype, whereas its genetic constitution is called its genotype. (Thus, 2 organisms with respective genotypes \(a A, b B, c c, d D, e e\) and \(A A, B B, c c,\) \(D D, e e\) would have different genotypes but the same phenotype. In a mating between 2 organisms, each one contributes, at random, one of its gene pairs of each type. The 5 contributions of an organism (one of each of the 5 types) are assumed to be independent and are also independent of the contributions of the organism's mate. In a mating between organisms having genotypes \(a A, b B, c C\) \(d D, e E\) and \(a a, b B, c c, D d, e e\) what is the probability that the progeny will (i) phenotypically and (ii) genotypically resemble (a) the first parent? (b) the second parent? (c) either parent? (d) neither parent?

Ninety-eight percent of all babies survive delivery. However, 15 percent of all births involve Cesarean (C) sections, and when a C section is performed, the baby survives 96 percent of the time. If a randomly chosen pregnant woman does not have a C section, what is the probability that her baby survives?

\(A\) and \(B\) are involved in a duel. The rules of the duel are that they are to pick up their guns and shoot at each other simultaneously. If one or both are hit, then the duel is over. If both shots miss, then they repeat the process. Suppose that the results of the shots are independent and that each shot of \(A\) will hit \(B\) with probability \(p_{A},\) and each shot of \(B\) will hit \(A\) with probability \(p_{B}\). What is (a) the probability that \(A\) is not hit? (b) the probability that both duelists are hit? (c) the probability that the duel ends after the \(n\) th round of shots? (d) the conditional probability that the duel ends after the \(n\) th round of shots given that \(A\) is not hit? (e) the conditional probability that the duel ends after the nth round of shots given that both duelists are hit?

Independent flips of a coin that lands on heads with probability \(p\) are made. What is the probability that the first four outcomes are (a) \(H, H, H, H ?\) (b) \(T, H, H, H ?\) (c) What is the probability that the pattern \(T, H, H, H\) occurs before the pattern \(H, H, H, H ?\) Hint for part \((c):\) How can the pattern \(H, H, H, H\) occur first?
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