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Given P(A) = 1/3, P(B) = 2/7, and P(A and B) = 3/7, find the probability of P(A or B).

Mathematics

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Answer

421\frac{4}{21}

Step-by-step solution

4 steps

  1. Identify the given probabilities.

    We are given the following probabilities:
    The probability of outcome A, P(A)=13P(A) = \frac{1}{3}.
    The probability of outcome B, P(B)=27P(B) = \frac{2}{7}.
    The probability of both A and B occurring, P(A∩B)=37P(A \cap B) = \frac{3}{7}.

  2. State the formula for the probability of either A or B.

    The probability of either A or B occurring, denoted as P(A∪B)P(A \cup B), is given by the formula:

    P(A∪B)=P(A)+P(B)−P(A∩B)P(A \cup B) = P(A) + P(B) - P(A \cap B)

  3. Substitute the given values into the formula.

    Substitute the values from Step 1 into the formula from Step 2:

    P(A∪B)=13+27−37P(A \cup B) = \frac{1}{3} + \frac{2}{7} - \frac{3}{7}

  4. Calculate the result.

    To add and subtract these fractions, find a common denominator, which is 21.

    P(A∪B)=1×73×7+2×37×3−3×37×3P(A \cup B) = \frac{1 \times 7}{3 \times 7} + \frac{2 \times 3}{7 \times 3} - \frac{3 \times 3}{7 \times 3} P(A∪B)=721+621−921P(A \cup B) = \frac{7}{21} + \frac{6}{21} - \frac{9}{21} P(A∪B)=7+6−921P(A \cup B) = \frac{7 + 6 - 9}{21} P(A∪B)=13−921P(A \cup B) = \frac{13 - 9}{21} P(A∪B)=421P(A \cup B) = \frac{4}{21}

    The probability of either A or B is 421\frac{4}{21}.

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