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Chapter 4: Problem 25

A hat contains 40 marbles. Of them, 18 are red and 22 are green. If one marble is randomly selected out of this hat, what is the probability that this marble is \(\begin{array}{ll}\text { a. red? } & \text { b. green? }\end{array}\)

Short Answer

Expert verified
The probability of picking a red marble is \( \frac{9}{20} \) , while picking a green marble is \( \frac{11}{20} \) .

Step by step solution

01

Determining the Probability of Picking a Red Marble

The probability of picking a red marble is calculated by dividing the number of red marbles (18) by the total number of marbles (40). Writing down the equation, \( P(Red) = \frac{18}{40} \) .
02

Simplifying the Probability of Picking a Red Marble

The above equation can be simplified by reducing the fraction. After simplifying, \( P(Red) = \frac{9}{20} \) .
03

Determining the Probability of Picking a Green Marble

Just like in step 1, calculate the probability of picking a green marble by using the equation \( P(Green) = \frac{22}{40} \) .
04

Simplifying the Probability of Picking a Green Marble

When we simplify the equation, we obtain \( P(Green) = \frac{11}{20} \) .

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Key Concepts

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

Random Selection
When we talk about probability and random selection, it means choosing an item without any bias. In our example, selecting a marble from a hat means each marble has an equal chance of being picked if selected randomly. This eliminates any favoritism, ensuring that each marble鈥攔egardless of its color鈥攈as the same probability of being selected. Since the process is random, every single choice is an independent event, unaffected by previous draws if you replace the marble each time. If you understand that each marble in the hat is equally likely to be chosen, you've grasped the core idea of random selection.
Simplifying Fractions
Fractions often represent relationships between numbers鈥攍ike our probabilities. Simplifying fractions means reducing them to their simplest form by dividing the numerator and the denominator by their greatest common divisor (GCD).
  • Take the probability of selecting a red marble: originally represented as \(\frac{18}{40}\).
  • To simplify, find the GCD of 18 and 40, which is 2.
  • Then, divide both 18 and 40 by 2, resulting in \(\frac{9}{20}\).
This simplified form is easier to understand and use in further calculations or comparisons. Simplifying fractions makes them neater and often easier to interpret, especially in mathematical or real-world applications.
Basic Probability Concepts
Probability is like a measure of how likely an event is to occur. It's expressed as a fraction, decimal, or percentage. To find the probability of a particular event, you compare the number of successful outcomes to the total possible outcomes.
  • Consider the probability of drawing a green marble. The probability formula is \(\frac{\text{Number of green marbles}}{\text{Total number of marbles}} = \frac{22}{40}\). Simplifying gives us \(\frac{11}{20}\).
  • If probabilities were decimals, they would lie between 0 and 1, where 0 means impossible and 1 means certain. The same concept in percentage is between 0% and 100%.
Probabilities closer to 1 (or 100%) indicate a higher likelihood of an event occurring, while those near 0 suggest it's less likely. Mastering these basic concepts helps in understanding and interpreting real-life occurrences and predictions.

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Most popular questions from this chapter

A statistical experiment has 10 equally likely outcomes that are denoted by \(1,2,3,4,5,6,7,8,9\), and 10 . Let event \(A=\\{3,4,6,9\\}\) and event \(B=\\{1,2,5\\}\). a. Are events \(A\) and \(B\) mutually exclusive events? b. Are events \(A\) and \(B\) independent events? c. What are the complements of events \(A\) and \(B\), respectively, and their probabilities?

A screening test for a certain disease is prone to giving false positives or false negatives. If a patient being tested has the disease, the probability that the test indicates a (false) negative is .13. If the patient does not have the disease, the probability that the test indicates a (false) positive is .10. Assume that \(3 \%\) of the patients being tested actually have the disease. Suppose that one patient is chosen at random and tested. Find the probability that a. this patient has the disease and tests positive b. this patient does not have the disease and tests positive c. this patient tests positive d. this patient has the disease given that he or she tests positive

A company hired 30 new college graduates last week. Of these, 16 are female and 11 are business majors. Of the 16 females, 9 are business majors. Are the events "female" and "business major" independ. ent? Are they mutually exclusive? Explain why or why not.

Five hundred employees were selected from a city's large private companies and asked whether or not they have any retirement benefits provided by their companies. Based on this information, the following two-way classification table was prepared. $$\begin{array}{llc} \hline & \text { Yes } & \text { No } \\ \hline \text { Men } & 225 & 75 \\ \text { Women } & 150 & 50 \\ \hline \end{array}$$ a. Suppose one employee is selected at random from these 500 employees. Find the following probabilities. i. Probability of the intersection of events "woman" and "yes" ii. Probability of the intersection of events "no" and "man" b. Mention what other joint probabilities you can calculate for this table and then find them. You may draw a tree diagram to find these probabilities.

Let \(A\) be the event that a number less than 3 is obtained if we roll a die once. What is the probability of \(A ?\) What is the complementary event of \(A\), and what is its probability?
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