Math Problem Statement

You choose a tile at random from a bag containing 2 ​A's, 2 ​B's, and 5 ​C's. You replace the first tile in the bag and then choose again. Find ​P(Upper C and Upper C​).

Solution

To solve this problem, we are asked to find the probability of selecting two CC's in a row from a bag containing:

  • 2 tiles labeled AA
  • 2 tiles labeled BB
  • 5 tiles labeled CC

Step 1: Total number of tiles in the bag

The total number of tiles is: 2+2+5=9 tiles in total.2 + 2 + 5 = 9 \text{ tiles in total.}

Step 2: Probability of choosing CC on the first draw

The probability of drawing a CC on the first draw is the ratio of CC's to the total number of tiles: P(C on first draw)=59P(\text{C on first draw}) = \frac{5}{9}

Step 3: Replacing the tile and drawing again

Since the tile is replaced after the first draw, the total number of tiles remains 9 for the second draw. The probability of drawing a CC again is: P(C on second draw)=59P(\text{C on second draw}) = \frac{5}{9}

Step 4: Calculating the probability of drawing CC on both draws

Since the draws are independent (the tile is replaced), the probability of drawing CC on both draws is the product of the individual probabilities: P(C and C)=P(C on first draw)×P(C on second draw)=59×59=2581P(\text{C and C}) = P(\text{C on first draw}) \times P(\text{C on second draw}) = \frac{5}{9} \times \frac{5}{9} = \frac{25}{81}

Final Answer:

P(C and C)=2581P(\text{C and C}) = \frac{25}{81}


Would you like more details or have any questions? Here are 5 related questions you might find useful:

  1. What would the probability be if there were only 3 CC's instead of 5?
  2. How would the result change if there was no replacement after the first draw?
  3. How do probabilities change if the number of AA's and BB's were different?
  4. What is the probability of drawing at least one CC in two draws?
  5. How can the concept of replacement influence other probability problems?

Tip: When calculating probabilities with replacement, remember that the total number of outcomes remains the same between draws.

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Math Problem Analysis

Mathematical Concepts

Probability
Independent Events
Multiplication Rule for Probability

Formulas

P(A and B) = P(A) * P(B)

Theorems

Multiplication Rule for Independent Events

Suitable Grade Level

Grades 6-8