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Chapter 17: Problem 3

Give the probability, as a percentage, of picking the indicated card from a deck. Red card

Short Answer

Expert verified
Answer: The probability of picking a red card from a standard deck of playing cards is 50%.

Step by step solution

01

Identify the number of red cards in the deck

In a standard deck of 52 playing cards, there are two red suits 鈥� hearts and diamonds. Each suit contains 13 cards, thus making a total of 26 red cards. So, the number of red cards = 13 (hearts) + 13 (diamonds) = 26 cards.
02

Calculate the ratio of red cards to the total number of cards in the deck

We need to find the ratio of red cards to the total number of cards. The ratio can be written as the fraction: Red cards / Total number of cards In our case, the ratio would look like this: 26 red cards / 52 total cards
03

Simplify the ratio/fraction if possible

Since both the numerator (red cards) and denominator (total cards) are even numbers, we can simplify the fraction. Dividing both by 2, we get: (26/2) / (52/2) 13 red cards / 26 total cards
04

Convert the ratio to a percentage

To convert the ratio or fraction into a percentage, we multiply the result by 100. (13 red cards / 26 total cards) * 100 0.5 * 100 = 50% The probability of picking a red card from a deck is 50%.

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Key Concepts

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

Playing Cards
Playing cards are a fascinating tool not just for games but also for understanding probability. A standard deck of playing cards consists of 52 individual cards. These cards are divided equally into four suits: hearts, diamonds, clubs, and spades.
  • The hearts and diamonds are referred to as the red suits, each consisting of 13 cards.
  • The clubs and spades are called black suits, also with 13 cards each.
The richness in variety of cards allows us to explore many probability scenarios simply and effectively. Since each suit has the same number of cards, it helps us maintain an equal distribution for various probability calculations.
Ratios
A ratio is a mathematical expression that compares two numbers or quantities. In probability, ratios are often used to compare the number of favorable outcomes to the total number of possible outcomes. For example, when we want to determine the probability of drawing a red card from a deck, we start by writing this as a ratio:
  • Number of red cards (favorable outcomes): 26
  • Total number of cards in the deck (possible outcomes): 52
This gives us the ratio \( \frac{26}{52} \), indicating that out of every 52 cards, 26 are red. We can simplify this ratio by finding the greatest common divisor, which is 2 in this case, reducing it to \( \frac{13}{26} \). Ratios provide a foundational step in understanding and conveying probability.
Percentages
Percentages are a common way to represent probability, often making the data easier to understand at a glance. It tells us how many parts of a hundred represent our interest. Converting a probability ratio to a percentage involves a simple calculation. Taking the simplified ratio from above, \( \frac{13}{26} \), the conversion to percentage is done by multiplying the ratio by 100.
  • \( \frac{13}{26} \times 100 = 50\% \)
Thus, there is a 50% probability of picking a red card from a deck, meaning out of any 10 times, you can expect to pick a red card 5 times. Percentages translate probabilities into a more intuitive and relatable format, making it easier to envisage likelihood in daily life.

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Most popular questions from this chapter

Find the mean and standard deviation of the data set. $$ 12,-8,13,-15,60,-72,23,-13 $$

Find the mean of the data set. $$ -5,-2,0,5,2 $$

A sample of \(n\) frogs has a total weight of \(W\) grams. (a) What is the mean weight of a frog in the sample? (b) The largest frog eats the smallest one. What is the mean weight now?

Table 17.19 gives the vehicle occupancy for people driving to work in 1990 as determined by the US Census. For instance, 84,215,000 people drove alone and 12,078,000 people drove in 2 -person car pools. Picking at random, what is the probability, given as a percentage, that: (a) A commuter drives to work alone? (b) A vehicle carries 4 or more people?$$ \begin{array}{c|c|c|c|c|c|c|c} \hline \text { Occupancy } & 1 & 2 & 3 & 4 & 5 & 6 & 7+ \\ \hline \text { People, } 1000 \mathrm{~s} & 84,215 & 12,078 & 2,001 & 702 & 209 & 97 & 290 \\ \hline \end{array} $$

The probability expressions refer to drawing a card from a standard deck of cards. State in words the meaning of the expression and give the probability as a fraction. \(P(\) Red \(\mid\) Heart \()\)
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