9. you draw three cards from a deck of cards. explain the difference in the probability of drawing three…

Explanation:

Step1: Calculate probability with replacement

The probability of drawing a king in a single - draw from a standard deck of 52 cards is $\frac{4}{52}=\frac{1}{13}$. Since the draws are independent when we replace the card, the probability of drawing three kings with replacement is $P_{1}=\left(\frac{4}{52}\right)\times\left(\frac{4}{52}\right)\times\left(\frac{4}{52}\right)=\left(\frac{1}{13}\right)^3=\frac{1}{2197}$.

Step2: Calculate probability without replacement

The probability of drawing the first king is $\frac{4}{52}$. The probability of drawing the second king without replacement is $\frac{3}{51}$ (as there are 3 kings left and 51 cards left). The probability of drawing the third king without replacement is $\frac{2}{50}$. So the probability of drawing three kings without replacement is $P_{2}=\frac{4}{52}\times\frac{3}{51}\times\frac{2}{50}=\frac{24}{132600}=\frac{1}{5525}$.

Step3: Find the difference

The difference in probabilities $\Delta P = P_{1}-P_{2}=\frac{1}{2197}-\frac{1}{5525}=\frac{5525 - 2197}{2197\times5525}=\frac{3328}{12132425}\approx0.000274$. The probability of drawing three kings is higher when the card is replaced because each draw is an independent event with the same probability of getting a king, while without replacement, the probability of getting a king in subsequent draws decreases as the number of kings and total cards in the deck decrease.

Answer:

The probability of drawing three kings with replacement is $\frac{1}{2197}$, and without replacement is $\frac{1}{5525}$. The difference is $\frac{3328}{12132425}\approx0.000274$, and the probability is higher when the card is replaced.