Math Problem Statement

Use the method of your choice to determine the following probability. Drawing at least one tenten when you draw a card from a standard deck 3 times​ (replacing the card each time you​ draw) Question content area bottom Part 1 The probability is enter your response here. ​(Round to the nearest thousandth as​ needed.)

Solution

To determine the probability of drawing at least one "tenten" (which I'll assume refers to a 10 in a standard deck), we can use the complement rule, which is often easier in these types of problems.

Step-by-step approach:

  1. Identify the probability of not drawing a 10:
    In a standard deck of 52 cards, there are 4 tens (one for each suit). So, the probability of not drawing a 10 is: P(not a ten)=52452=4852=1213P(\text{not a ten}) = \frac{52 - 4}{52} = \frac{48}{52} = \frac{12}{13}

  2. Use complement rule for at least one event:
    The probability of drawing at least one ten in 3 draws (with replacement) is the complement of drawing no tens in all 3 draws.

    The probability of not drawing a 10 in a single draw is 1213\frac{12}{13}. Therefore, the probability of not drawing a 10 in 3 consecutive draws is: P(no ten in 3 draws)=(1213)3P(\text{no ten in 3 draws}) = \left( \frac{12}{13} \right)^3 We can compute this as: P(no ten in 3 draws)=(1213)3=1213×1213×1213P(\text{no ten in 3 draws}) = \left( \frac{12}{13} \right)^3 = \frac{12}{13} \times \frac{12}{13} \times \frac{12}{13}

  3. Subtract from 1 to get the complement:
    The probability of drawing at least one 10 in 3 draws is: P(at least one ten)=1P(no ten in 3 draws)P(\text{at least one ten}) = 1 - P(\text{no ten in 3 draws}) So, let's compute this value.

Now, I'll calculate these values.

The probability of drawing at least one ten in 3 draws (with replacement) is approximately 0.213 (rounded to the nearest thousandth).

Would you like further details or have any questions?

Here are 5 related questions to explore:

  1. What is the probability of drawing exactly one ten in 3 draws with replacement?
  2. How does the probability change if you draw without replacement?
  3. What is the probability of drawing at least two tens in 3 draws with replacement?
  4. What is the probability of drawing no tens in 5 draws with replacement?
  5. How would the probability change if there were only 2 tens in the deck?

Tip: In probability, using the complement rule is often simpler for "at least one" scenarios.

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Math Problem Analysis

Mathematical Concepts

Probability
Complement Rule

Formulas

P(not a 10) = (52 - 4) / 52 = 48 / 52 = 12 / 13
P(no 10 in 3 draws) = (12/13)^3
P(at least one 10) = 1 - P(no 10 in 3 draws)

Theorems

Complement Rule

Suitable Grade Level

Grades 9-11