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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

Two thousand randomly selected adults were asked whether or not they have ever shopped on the Internet. The following table gives a two-way classification of the responses. $$\begin{array}{lcc} \hline & \text { Have Shopped } & \text { Have Never Shopped } \\ \hline \text { Male } & 500 & 700 \\ \text { Female } & 300 & 500 \end{array}4$ a. If one adult is selected at random from these 2000 adults, find the probability that this adult i. has never shopped on the Internet ii. is a male iii. has shopped on the Internet given that this adult is a female iv. is a male given that this adult has never shopped on the Internet b. Are the events "male" and "female" mutually exclusive? What about the events "have shopped" and "male?" Why or why not? c. Are the events "female" and "have shopped" independent? Why or why not?

A random sample of 2000 adults showed that 1320 of them have shopped at least once on the Internet. What is the (approximate) probability that a randomly selected adult has shopped on the Internet?

A player plays a roulette game in a casino by betting on a single number each time. Because the wheel has 38 numbers, the probability that the player will win in a single play is \(1 / 38\). Note that each play of the game is independent of all previous plays a. Find the probability that the player will win for the first time on the 10 th play. b. Find the probability that it takes the player more than 50 plays to win for the first time c. The gambler claims that because he has 1 chance in 38 of winning each time he plays, he is certain to win at least once if he plays 38 times. Does this sound reasonable to you? Find the probability that he will win at least once in 38 plays

A gambler has given you two jars and 20 marbles. Of these 20 marbles, 10 are red and 10 are green You must put all 20 marbles in these two jars in such a way that each jar must have at least one marble in it. Then a friend of yours, who is blindfolded, will select one of the two jars at random and then will randomly select a marble from this jar. If the selected marble is red, you and your friend win \(\$ 100\) a. If you put 5 red marbles and 5 green marbles in each jar, what is the probability that your friend selects a red marble? b. If you put 2 red marbles and 2 green marbles in one jar and the remaining marbles in the other jar, what is the probability that your friend selects a red marble? c. How should these 20 marbles be distributed among the two jars in order to give your friend the highest possible probability of selecting a red marble?

Given that \(A\) and \(B\) are two independent events, find their joint probability for the following. a. \(P(A)=.61\) and \(P(B)=.27\) b. \(P(A)=.39\) and \(P(B)=.63\)
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