Math Problem Statement

solve part b

Solve part b of the given problem.

Solution

Let's break down Part (b) of the question into manageable parts and solve step by step.


(i) Describe the incident AA in words:

The sample space grid is shown, and incident AA corresponds to the diagonal line where the two numbers picked are the same.
Thus, AA represents the event:

"The first card and second card have the same number."

(ii) Find the probability of getting AA:

The sample space consists of all possible outcomes when two cards are picked (with replacement). Since each card has 5 numbers (1,2,3,4,51, 2, 3, 4, 5), the total sample space is: 5×5=25.5 \times 5 = 25.

The favorable outcomes for AA (same number on both cards) are: (1,1),(2,2),(3,3),(4,4),(5,5).(1, 1), (2, 2), (3, 3), (4, 4), (5, 5).

Thus, there are 5 favorable outcomes. The probability of AA is: P(A)=Number of favorable outcomesTotal outcomes=525=15.P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{5}{25} = \frac{1}{5}.


(iii) If event BB is "sum of both numbers on the card is 8\geq 8," mark BB on the grid and find P(B)P(B):

Step 1: Find outcomes where the sum 8\geq 8

Let the numbers from the first and second cards be xx and yy, respectively. The sum x+y8x + y \geq 8 includes the following pairs:

  • For x=5x = 5: y=3,4,5y = 3, 4, 5(5,3),(5,4),(5,5)(5, 3), (5, 4), (5, 5)
  • For x=4x = 4: y=4,5y = 4, 5(4,4),(4,5)(4, 4), (4, 5)
  • For x=3x = 3: y=5y = 5(3,5)(3, 5)

Thus, the favorable outcomes are: (5,3),(5,4),(5,5),(4,4),(4,5),(3,5).(5, 3), (5, 4), (5, 5), (4, 4), (4, 5), (3, 5).

Step 2: Count the favorable outcomes

There are 6 outcomes where x+y8x + y \geq 8.

Step 3: Find the probability of BB

The probability of BB is: P(B)=Number of favorable outcomesTotal outcomes=625.P(B) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{6}{25}.


Final Answers:

  1. (i) Event AA: "The first and second cards have the same number."
  2. (ii) P(A)=15P(A) = \frac{1}{5}.
  3. (iii) P(B)=625P(B) = \frac{6}{25}.

Would you like me to further explain any part of the solution or the steps involved? Here are 5 related questions you might find interesting:

  1. How do we calculate probabilities for dependent events?
  2. Can we visualize outcomes using other tools like tables?
  3. How would the probabilities change if the cards were not replaced?
  4. What happens if the numbers on the cards ranged from 1 to 10 instead?
  5. How do we define complementary probabilities for events like AA and BB?

Tip: For problems with grids or large sample spaces, systematically count favorable outcomes to ensure accuracy.

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Math Problem Analysis

Mathematical Concepts

Probability
Sample Space
Event Analysis

Formulas

P(Event) = Number of Favorable Outcomes / Total Number of Outcomes
Addition of Numbers (Sum >= 8)

Theorems

Basic Probability Theorem

Suitable Grade Level

Grade 10-12