Example Question - deck of cards
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Differentiation of xcosx and Probability of Drawing Specific Cards from a Deck
<p>The image displays two separate questions. I will provide the solutions for both.</p> <p>For the differentiation of \( x\cos{x} \) with respect to \( x \) using the first principle:</p> <p>We have \( f(x) = x\cos{x} \), we need to find \( f'(x) \) using the first principle:</p> <p>\[ f'(x) = \lim_{{h \to 0}} \frac{f(x+h) - f(x)}{h} \]</p> <p>\[ = \lim_{{h \to 0}} \frac{(x+h)\cos(x+h) - x\cos{x}}{h} \]</p> <p>We then expand and arrange the expression and apply the limit. However, without the options or further context for finding the value of \( k \), this part of the question is incomplete.</p> <p>For the probability question, assuming we are looking to find the probability of drawing 1 diamond and 3 spades:</p> <p>The total number of ways to draw 4 cards from a 52 card deck is \( C(52, 4) \).</p> <p>The number of ways to draw 1 diamond from the 13 available diamonds is \( C(13, 1) \).</p> <p>The number of ways to draw 3 spades from the 13 available spades is \( C(13, 3) \).</p> <p>The probability \( P \) of the event is:</p> <p>\[ P = \frac{C(13, 1) \cdot C(13, 3)}{C(52, 4)} \]</p> <p>\[ P = \frac{13 \cdot \frac{13!}{3!(13-3)!}}{\frac{52!}{4!(52-4)!}} \]</p> <p>We can then simplify the factorials to get the probability.</p>
Probability of Drawing Cards from a Deck
<p>Let E be the event that the card drawn is not a heart.</p> <p>There are a total of 52 cards in a deck, and there are 13 cards of each suit, including hearts.</p> <p>\( P(E) = \frac{Number\ of\ non-heart\ cards}{Total\ number\ of\ cards} \)</p> <p>\( P(E) = \frac{52 - 13}{52} \)</p> <p>\( P(E) = \frac{39}{52} \)</p> <p>\( P(E) = \frac{3}{4} \)</p> <p>So, the probability that the card drawn is not a heart is \(\frac{3}{4}\).</p>
Card Drawing Probability Calculation
The problem states that Ming has a standard 52-card deck, which consists of 4 suits with an equal number of cards in each suit. Ming is going to draw a card from the deck 300 times, with the card being replaced in the deck after each draw. Since there are 4 suits and the deck is well-shuffled each time, the probability of drawing a card from any specific suit (such as a heart) is 1/4. To find out how many times, on average, Ming might draw something other than a heart in 300 draws, we calculate the expectation for the other suits by subtracting the probability of drawing a heart from 1: 1 - (1/4) = 3/4 So the probability of not drawing a heart on any given draw is 3/4. To calculate the expected number of times she will not draw a heart in 300 draws: (3/4) * 300 = 225 Ming will, on average, draw something other than a heart 225 times. Therefore, the best prediction to complete the statement is: "Ming will draw something other than a ♥️ exactly 225 times." So the answer is "Exactly 225 times."
Probability of Choosing Red Court Cards in a Deck of Cards
To solve this question, we need to understand what a red court card is and how many are in a standard deck of cards. In a standard deck, there are 52 cards. There are two red suits: diamonds and hearts. Each of these suits has three "court" or "face" cards: the Jack, Queen, and King. So for diamonds, there's a Jack, Queen, and King, and the same for hearts. That's a total of 6 red court cards. The probability of choosing one of these red court cards from a full deck is the number of red court cards divided by the total number of cards. Probability = Number of red court cards / Total number of cards in the deck Probability = 6 / 52 Simplify the fraction by dividing both the numerator and denominator by the greatest common divisor, which is 2 in this case. Probability = 3 / 26 So the probability of choosing a red court card from a deck of cards is 3 in 26, or approximately 11.54%.
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