14. A ball is drawn at random from a box containing 3 red balls, 2 white balls, and 5 blue balls. What is the probability of picking a red ball?
A. [tex]$1 / 3$[/tex]
B. [tex]$7 / 10$[/tex]
C. [tex]$2 / 5$[/tex]
D. [tex]$3 / 2$[/tex]
15. If x = -2, y = 4, and k = 5, evaluate 4xy + 7k
Answer (2)
Calculate the total number of balls: T o t a l = 3 + 2 + 5 = 10 .
Calculate the probability of picking a red ball: P ( R e d ) = 10 3 .
Evaluate the expression 4 x y + 7 k with x = − 2 , y = 4 , and k = 5 : 4 ( − 2 ) ( 4 ) + 7 ( 5 ) = − 32 + 35 = 3 .
The probability of picking a red ball is 10 3 and the value of the expression is 3 .
Explanation
Understand the problem We are given a box containing 3 red balls, 2 white balls, and 5 blue balls. We want to find the probability of picking a red ball when one ball is drawn at random.
Calculate the total number of balls First, we need to find the total number of balls in the box. We add the number of red, white, and blue balls: T o t a l = R e d + Whi t e + Bl u e = 3 + 2 + 5 = 10
Calculate the probability of picking a red ball Next, we calculate the probability of picking a red ball. The probability is the ratio of the number of red balls to the total number of balls: P ( R e d ) = T o t a l n u mb er o f ba ll s N u mb er o f R e d ba ll s = 10 3
Compare with the given options The probability of picking a red ball is 10 3 . Comparing this to the given options, we see that option A is 3 1 , option B is 10 7 , option C is 5 2 , and option D is 2 3 . Our calculated probability matches none of the options exactly, but it is closest to 10 3 . However, since we are asked to choose the correct option, and 10 3 is not listed, there might be an error in the question or the options. Assuming the question is correct, the probability of picking a red ball is 10 3 .
Evaluate the expression For the second part of the question, we are given x = − 2 , y = 4 , and k = 5 . We need to evaluate the expression 4 x y + 7 k .
4 x y + 7 k = 4 ( − 2 ) ( 4 ) + 7 ( 5 ) = − 32 + 35 = 3 Therefore, the value of the expression is 3.
Examples
Probability calculations are used in various real-life scenarios, such as predicting the likelihood of winning a lottery, determining the risk of an investment, or forecasting weather patterns. For example, if you know the probability of rain on a given day is 3 1 , you can decide whether or not to carry an umbrella. Similarly, in games of chance, understanding probabilities helps in making informed decisions. In this case, knowing the number of balls of each color in a box allows us to calculate the likelihood of drawing a specific color, which is a fundamental concept in probability.
The probability of picking a red ball from the box is 10 3 , and when evaluating the expression 4 x y + 7 k with given values, the result is 3 .
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