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A Venn diagram shows two intersecting circles. Circle 1 represents event A, 3 and 5. Circle 2 represents event B, 6, 8, 10, and 12. The intersection of both the circles is labeled 7, 9, and 11. The space outside the circles is labeled 2 and 4.

The Venn diagram represents the results of the experiment of rolling two fair, six-sided number cubes and adding the sum of the two rolls. Consider event A
as “sum is odd” and event B
as “sum is 6 or greater.” Find the probability that the sum is odd, given that the sum is 6 or greater.

(1 point)
Responses

37
Start Fraction 3 over 7 End Fraction

35
Start Fraction 3 over 5 End Fraction

312
Start Fraction 3 over 12 End Fraction

512
Start Fraction 5 over 12 End Fraction
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1 answer

To find the probability that the sum is odd (event A) given that the sum is 6 or greater (event B), we can use the conditional probability formula:

\[ P(A | B) = \frac{P(A \cap B)}{P(B)} \]

  1. Identify outcomes:

    • The possible sums when rolling two six-sided dice range from 2 to 12.
    • The sums listed in Circle 1 (event A: "sum is odd") are 3, 5, 7, 9, and 11.
    • The sums listed in Circle 2 (event B: "sum is 6 or greater") are 6, 7, 8, 9, 10, 11, and 12.
    • The intersection (A ∩ B) contains the sums that are both odd and 6 or greater, which are 7, 9, and 11.

  2. Count outcomes:

    • The favorable outcomes for \( A \cap B \) (sum is odd and 6 or greater) are 7, 9, and 11. This gives us 3 outcomes.
    • The outcomes for B (sums 6 or greater) are 6, 7, 8, 9, 10, 11, and 12. This gives us 7 outcomes.

  3. Calculate probabilities:

    • \( P(A \cap B) = \frac{3}{36} \) since there are 3 favorable outcomes out of 36 possible outcomes when rolling two dice.
    • \( P(B) = \frac{7}{36} \) since there are 7 favorable outcomes out of 36 possible outcomes.

  4. Use the conditional probability formula: \[ P(A | B) = \frac{P(A \cap B)}{P(B)} = \frac{\frac{3}{36}}{\frac{7}{36}} = \frac{3}{7} \]

Thus, the probability that the sum is odd given that the sum is 6 or greater is:

\[ \boxed{\frac{3}{7}} \]