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Introduction to Probability · Ages 15–16

Conditional Probability Practice

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Use these problems to practice reading conditions, shrinking the sample space, and calculating conditional probabilities. Write each answer as a fraction in simplest form.

  1. A standard six-sided die is rolled. Find the probability that the number is 6, given that the roll is an even number.

  2. A standard six-sided die is rolled. Find the probability that the number is greater than 3, given that the roll is an odd number.

  3. In a club of 30 members, 12 play chess, 10 play bridge, and 4 play both. Find P(Bridge | Chess).

  4. Using the same club data (12 play chess, 10 play bridge, 4 play both), find P(Chess | Bridge).

  5. A survey of 50 pets shows 30 dogs and 20 cats. Among the dogs, 18 have short fur. Among the cats, 14 have short fur. A pet is chosen at random and has short fur. What is the probability that it is a dog?

  6. Two coins are tossed. Find the probability that both are heads, given that at least one coin landed on heads.

  7. If P(A) = 0.5, P(B) = 0.4, and P(A and B) = 0.2, calculate P(B | A).

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Conditional Probability Practice

Answer key

  1. A standard six-sided die is rolled. Find the probability that the number is 6, given that the roll is an even number.

    1/3. The condition 'even' shrinks the sample space to {2, 4, 6}. Exactly one of these outcomes is 6.

  2. A standard six-sided die is rolled. Find the probability that the number is greater than 3, given that the roll is an odd number.

    1/3. The odd numbers are {1, 3, 5}. Only 5 is greater than 3, giving 1 out of 3.

  3. In a club of 30 members, 12 play chess, 10 play bridge, and 4 play both. Find P(Bridge | Chess).

    4/12 or 1/3. The condition is Chess (12 members). The intersection is 4 members.

  4. Using the same club data (12 play chess, 10 play bridge, 4 play both), find P(Chess | Bridge).

    4/10 or 2/5. The condition is Bridge (10 members). The intersection is 4 members.

  5. A survey of 50 pets shows 30 dogs and 20 cats. Among the dogs, 18 have short fur. Among the cats, 14 have short fur. A pet is chosen at random and has short fur. What is the probability that it is a dog?

    18/32 or 9/16. Total short-furred pets = 18 + 14 = 32. Dogs with short fur = 18.

  6. Two coins are tossed. Find the probability that both are heads, given that at least one coin landed on heads.

    1/3. The sample space shrinks to {HH, HT, TH}, of which only HH has both heads.

  7. If P(A) = 0.5, P(B) = 0.4, and P(A and B) = 0.2, calculate P(B | A).

    0.4 (or 2/5). P(B | A) = P(A and B) / P(A) = 0.2 / 0.5 = 0.4.